ACT Math Prep | Master counting, Probability and Geometry

Define probability as chance an event occurs, ranging from zero to one; calculate it as favorable over total, with examples like heads on a coin and even numbers on dice.

Compute the blue-ball probability by counting favorable events and total events in a box with 20 red and 10 blue balls, yielding a probability of 10/30 = 1/3 (33%).

Explore and or in probability using dice examples, visualize with Venn diagrams, and apply intersection, union, and independent events rules to calculate probabilities.

Calculate the probability of drawing blue balls from two boxes with independent draws, by multiplying the individual blue probabilities: 10/30 and 15/45, yielding 1/9.

Distribute twenty blue and twenty yellow balls into two boxes and pick one from each to maximize the probability of both being yellow.

Employ a three-circle Venn diagram for numbers 1–100, count multiples of 2, 5, and 7, apply inclusion-exclusion to get 66 favorable outcomes and probability 66/100.

Understand the complement of an event and apply P(E^c)=1-P(E) to compute probabilities. See how rolling a die with E defined as even numbers shows that P(E)+P(E^c)=1.

Practice a probability problem with two dice, finding P(sum ≤ 10) by complement: compute P(11) and P(12) as 2/36 and 1/36, yielding 11/12.

Master the concept of exhaustive events by recognizing sets that cover all outcomes, illustrated with dice examples and probability sums to one, including the relationship to an event's complement.

Apply the complement rule to find the probability that at least one of three people catches a fish, using their individual catch probabilities of 1/3, 1/4, and 3/7.

Explain mutually exclusive events with a coin toss, showing no overlap and that P(A or B) equals P(A) plus P(B); if not exclusive, subtract the intersection.

Calculate the probability of sums 9 or 10 when rolling two dice, treating them as mutually exclusive. There are four ways for 9 and three for 10, totaling 7/36.

Calculate the probability of an even sum or a multiple of five when rolling two dice, using the union and intersection of events in a 36-outcome sample space.

Explore independent events and compute joint probability using P(A)P(B) for independent cases; when not independent, multiply P(A) by P(B) given A.

Compare drawing with and without replacement from a box of red and blue balls: with replacement, red then blue is 12/49; without, 12/42.

Learn conditional probability by deriving P(E2|E) = P(E2 ∩ E) / P(E), using outcomes and events to compute probabilities.

Apply conditional probability to find the probability of newspaper two given newspaper one using P(B|A)=P(A∩B)/P(A), with 40% subscribing to newspaper one and 20% to both.

Use conditional probability to find the probability of absence given Monday. Compute P(absent and Monday) = 0.05 and P(Monday) = 0.2, yielding 0.25.

This lecture shows how to compute the probability of exactly three successes in five independent matches using the binomial coefficient (nCr) with p = 2/3 and q = 1/3.

Compute the probability that tails appear exactly three times in seven tosses of an unbiased coin using the binomial formula with p=1/2, yielding 35/128.

Compute the probability of the team’s fourth win on the 10th match by having exactly three wins in the first nine and winning the tenth, with p = 1/3.

Clarify odds in favor and odds against with a team example, showing that 1 to 3 yields a 1/4 probability of winning and 3/4 of losing; they are complementary.

Convert the odds in favor of winning as 1 to 3 into a win probability of 1/4, then apply the binomial formula for four wins in five games, yielding 15/1024.

Explore basic geometry concepts by examining points, line segments, and lines, and learn how points determine locations, line segments have endpoints, and lines extend indefinitely.

Count line segments from five collinear points by selecting two endpoints; use 5 choose 2 equals 10 and verify with a stepwise count from A through D.

Define intersecting lines as lines sharing a single point, the point of intersection, and that two lines cannot share more than one point. Parallel lines on a plane never meet.

Practice true-or-false geometry problems about lines: analyze whether multiple lines can pass through a single point and whether two points determine a unique line.

Explore the concept of a plane as a flat, two-dimensional surface with length and width, and distinguish it from one-dimensional lines using everyday examples like screens and tables.

Learn how a ray is a portion of a line that starts at a point and extends infinitely in one direction, with naming conventions like AB representing the ray.

Study polygons as simple closed figures made of line segments, including pentagons, quadrilaterals, and triangles; learn naming by ordered vertices and that at least three sides are required.

Explains basic polygon terms using a pentagon, naming corners A, B, C, D, E, identifying sides, vertices, adjacent vertices, and five diagonals.

Explore diagonals in polygons, noting that triangles have zero diagonals and a hexagon has nine using the formula nC2 minus n. The lesson shows counting diagonals by connecting non-adjacent vertices.

An angle is formed by two rays with a common endpoint, the vertex. We indicate it with a curve and name it using three points, e.g., angle BEC.

Learn to measure angles using degrees, use a protractor, identify right, straight, and complete angles, and classify acute, obtuse, and reflex angles with examples.

Explore related angles by identifying complementary and supplementary pairs, adjacent angles and linear pairs that sum to 180 degrees, and vertically opposite angles that are equal.

Practice problem explains complementary and supplementary angles on a straight line. With angle two at 90 degrees, angle one plus angle three equals 90, and total equals 180 degrees.

Explore angles formed by compass directions, showing that the angle between northwest and northeast is 90 degrees and the angle between south and east is 90 degrees, making them supplementary.

Determine that the vertical opposite angles are equal to 100 degrees, then use the straight-angle sum of 180 degrees to solve for B as 20 degrees.

Evaluate whether two angles are at different angles by checking a common vertex, a common arm, and arms on either side of the common arm.

Identify a transversal as a line that intersects two or more lines at distinct points. If the intersections coincide, the line is not a transversal.

Identify which statement is false about angles formed by intersecting lines, recognizing that vertically opposite angles are equal, linear pairs sum to 180, and straight angles measure 180 degrees.

Explore how a transversal creates eight angles with two lines, classifying interior, exterior, corresponding, alternate interior, and alternate exterior angles, and identify same-side interior pairs.

Analyze how a transversal across parallel lines yields equal corresponding, alternate interior, and alternate exterior angles, with interior angles on the same side supplementary, and vertical opposite angles linking relations.

Learn to identify parallel lines using a transversal by checking corresponding, alternate interior or alternate exterior angles, or consecutive interior angles that sum to 180 degrees.

Apply parallel line reasoning in a w-shaped figure, using BQ and QR as parallels with a transversal to identify alternate interior angles, concluding X and Y both equal 39 degrees.

Determine which lines are parallel in the given figure by testing angles with a transversal. Conclude that ef and gh are parallel, while other line pairs are not.

Practice problem on parallel lines and angle relationships shows how to find angle QPR using alternating angles, linear pair, and triangle sum; the answer is 80 degrees.

Apply parallel lines and a transversal to a geometry problem, use alternate interior angles to find A1 = 60° and A2 = 30°, then A1 + A2 = 90°.

Explore triangles by identifying sides, angles, and vertices in triangle ABC, and apply the property that the angle opposite the greater side is larger.

Explore classification of triangles by sides and by angles, including equilateral, isosceles, and scalene triangles, and acute, right, and obtuse angle triangles with their defining properties.

Classify triangles by sides and angles to reinforce fundamentals, identifying isosceles, scaling triangle, equilateral triangles, and acute, right, obtuse angle triangles through practice.

Master the exterior angle property of a triangle, proving that an exterior angle equals the sum of its interior opposite angles, with parallel lines and alternate interior angles.

Apply the exterior angle theorem in triangles to find an exterior angle by summing interior opposite angles, demonstrated with 60 and 40 degrees yielding 100 degrees.

Explore the triangle angle sum property, showing that the interior angles sum to 180 degrees and that an exterior angle equals the sum of the opposite interior angles.

Showcases that the sum of exterior angles in order around a triangle equals 360 degrees, and extends this to any polygon using interior angle sums and turning angles.

Apply the triangle angle sum property to find angle one as 70 degrees, then use the straight angle with 90 degrees to get x = 20 degrees.

Apply the external angle theorem to relate the exterior angle to interior angles, deduce 60 and 45 degrees, then use straight- and triangle-angle sums to find X equals 90.

Apply the triangle angle sum and the angle bisectors of the other two angles to show angle three equals 145 degrees.

Apply isosceles triangle properties and triangle angle sums to chase angles, use the exterior angle theorem, and find angle e equals 105 degrees.

Explore congruence of triangles by learning how two triangles become exact copies when superimposed, with corresponding vertices, sides, and angles. Apply four checks: sss, sas, asa, rhs to determine congruence.

Practice congruent triangles by orienting triangle XYZ to match the given triangle, using 3.5 cm and 5 cm sides with a 45-degree angle, then identify corresponding vertices.

Use the SSS criteria to determine triangle congruence by comparing three corresponding sides; if they match, the triangles are congruent and a unique copy can be drawn.

Explore a triangle congruence problem on the same base: AC equals DB and BC is common, with three equal side pairs, proving ABC is congruent to DCB by SSS.

The sas congruence criterion is explained, showing that two sides and the included angle define a congruent triangle, demonstrated by constructing a copy of triangle ABC.

Explore the asa and aas criteria for triangle congruency: two angles with the included side (asa) or two angles with any side (aas) ensure congruent triangles.

Apply the RHS criteria to check congruence of right-angled triangles by matching equal hypotenuse and one corresponding side, then construct a copy from the given angle and sides.

Show how midpoints BM = ME and AM = MC form congruent triangles, use vertical angles, and conclude equal corresponding angles to determine option B.

Practice problem on congruent triangles helps you identify corresponding sides. Given PQ = 5 cm, QR = 6 cm, RP = 7 cm, you find TU = 6 cm.

Prove triangles QSR and ART are congruent as right triangles using the angle–hypotenuse–side criterion with a common side QR. Conclude corresponding angles are equal.

Identify common mistakes in triangle congruency by examining four rules for checking two triangles, and show why SSA is not sufficient to guarantee congruency, using equilateral examples.

Explore triangle congruency criteria, identify why SSA is not valid, and use equilateral triangles to show why equal angles alone cannot prove congruence.

Demonstrate Pythagoras's theorem: in a right angle triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, via rearranged triangles and area arguments.

Learn how Pythagorean triplets work, with five, three, and four as an example, and see that multiplying a triplet by any number produces another valid triplet.

Practice identifying Pythagorean triplets by checking sums of squares for options, and learn that multiplying a known triple by a constant yields another Pythagorean triplet.

Apply pythagoras to a broken tree problem, find the hypotenuse from 5 m and 12 m, then add to determine the actual height of 18 m.

Compute the distance between the tops of two verticals 7 m and 4 m tall, 4 m apart, using a right triangle and Pythagoras to get 5.

Determine the ship's speed from the lighthouse geometry. Eight km from the lighthouse with a 20 square root of two minutes interval gives 8√2 km, yielding 20/3 m/s after conversion.

Learn how the Pythagoras theorem classifies triangles by comparing B^2 to A^2 + C^2: right (equal), acute (less), obtuse (greater), with a 5–10 example showing obtuse.

Learn to compute the area of a triangle as half the area of a parallelogram, using base times height as the formula.

Compute the triangle's third side given base 20 cm, side 15 cm, and area 90 cm² by using height and base–height area, then apply Pythagoras for about 12 cm.

Explore how triangles between parallel lines with the same base have equal areas, since area equals half the base times the height.

Apply a quick geometric construction in a rectangle to form six equal-area triangles between parallel lines, then deduce the area ratio of the triangle pq d to dq is 4 to 1.

Similar triangles are shapes with the same form but not the same size, where matching angles are equal and the ratios of matching sides are equal.

Identify triangle similarity using two criteria: if two angles match, the triangles are similar; or if two side ratios are equal with the included angle equal.

Use parallel lines and similar triangles in a circle to relate segments, derive AC = 16 cm from the area 64 cm², then compute DG = DC = 2√2.

Explore how similarity in triangles lets you compare areas: the ratio of areas equals the square of the ratio of corresponding sides, using base and height reasoning.

Apply similarity of triangles apq and abc with bq parallel to bc; from area ratio 1:4, ap/ab = 1/2, giving ap = 5 cm and bb = 5 cm.

Explore the three altitudes of a triangle by dropping perpendiculars from each vertex to the opposite side, defining height as the perpendicular distance, with inside and outside examples.

Explore the altitudes of triangle ABC: a triangle has three altitudes, and in obtuse triangles at least one altitude lies outside the triangle.

Identify how altitudes lie inside or outside a triangle: obtuse triangles create an exterior altitude, while a right triangle has all altitudes inside; the correct answer is option B.

Explore how the three altitudes of a triangle intersect at the orthocenter, the point where they meet.

Identify the midpoint of BC and construct perpendicular bisectors for all sides; their intersection is the circumcenter, yielding the circumcircle with radius equal to the circumradius.

Explore how angle bisectors meet at a triangle's center to draw the inscribed circle, or inner circle, tangent to all three sides.

Explore medians of a triangle, lines from vertices to midpoints of opposite sides, and how the centroid divides each median at 2:1, creating two equal-area triangles.

Identify the median to the hypotenuse in a right triangle; midpoint of ac is circle center with radii to a, b, c. If ab equals bc, the triangle is isosceles.

Solve a right triangle problem with the shortest median 15 cm and area 216 cm²; derive legs 18 and 24, then find the longest median as 3√73 cm.

Practice problem 2 reinforces triangle geometry by identifying the median from vertex B to the midpoint of AC and recognizing the altitude through C, FC.

Use the 15 cm radius of the equilateral triangle to find its side length, compute the area with sqrt(3)/4 side^2, and take three-fourths for quadrilateral abcde, yielding 2025 sqrt(3)/16 cm^2.

Understand isosceles triangles with two equal sides and equal opposite angles. See how the apex angle bisector to BC yields perpendicular bisector, median, and equal-area halves.

Solve an isosceles triangle problem: with one angle of 100 degrees, use the 180-degree triangle sum to find the two equal angles as 40 degrees each.

Learn about equilateral triangles, where all sides and angles are equal; derive the height as sqrt(3)/2 times the side and the area as sqrt(3)/4 times the side squared.

In an equilateral triangle, the median, angle bisector, perpendicular bisector, and altitude coincide; the centroid, circumcenter, and incenter also coincide, with circumradius two-thirds and inradius one-third of the altitude.

In this practice problem, an equilateral triangle sits inside a square, guiding you to use equal sides and angle sums to prove angle APQ equals 15 degrees.

Solve a geometry problem in an equilateral triangle using altitude as median, prove congruent right triangles, and apply Pythagoras to compare squared sides and identify the correct option.

Explore the Euler line of a triangle, connecting the orthocenter, centroid, and circumcenter; the centroid divides the orthocenter and circumcenter segment in a 2:1 ratio in non-equilateral triangles.

Explore how altitudes, medians, perpendicular bisectors, and angle bisectors converge at the triangle’s centers, namely orthocenter, centroid, circumcenter, and incenter, and align on the Euler line with circumradius and inradius.

Explore four formulas for triangle area: base–height, Heron's formula with semiperimeter s, area via inradius r s, and abc over 4R using circumradius.

Compute the semicircle radius in a triangle with a semicircle whose diameter lies on a side, using tangent-right-angle properties and area decomposition from sides 13, 4, and 15 cm.

Apply pythagorean triplets to identify right triangles with sides 15, 20, 25, then compute the inradius from area and semiperimeter and sum the radii to get 7 cm.

Learn a fast shortcut for right triangles using the 3-4-5 triplet and its multiples to find the inradius and circumradius, with area relationships to solve geometry questions quickly.

Solve a multi concept geometry problem on triangle park using tangents from an external point, inner circle, kite properties, and angle chasing to find angle q.

Understand triangle inequality: sum of any two sides exceeds the third, and the difference of two sides is less than the third; for 5 and 12, third side lies 8–16.

Apply the triangle inequality to test side lengths, where the sum of any two sides exceeds the third; 3,4,5 works, while 2,4,6 cannot.

Apply the triangle inequality to sides seven and nine cm to find the third side must lie between 3 and 15 cm, confirming option C.

This practice problem reinforces the triangle inequality: the sum of any two sides must exceed the third. Based on the checks, the correct answer is option D.

Explore solving a 10 cm equilateral triangle problem by dropping perpendiculars from an interior point, partitioning the triangle into sub-triangles, and deriving a+b+c = 5√3.

Master the interior angle bisector theorem: a bisected angle divides the opposite side into segments proportional to the adjacent sides, with correct numerator and denominator order.

Use the interior angle bisector theorem to relate BP, BQ, and PR under AR:ER = 2:1, with BQ = 3 cm and AC = 14 cm. BP equals 5 cm.

Explore the external angle bisector theorem in triangles, showing how the external angle yields a proportional relationship between adjacent sides by treating it as an internal angle.

Determine the triangle's sides as 20, 24, and 8 from the given ratios; apply Heron's formula to obtain area 12 sqrt(39), then compute inradius r = area/26.

Covers the midpoint theorem, showing the segment between midpoints is parallel to the base and half its length, then introduces the proportionality theorem and Apollonius theorem.

Learn the basics of polygons and quadrilaterals, including what makes a simple closed figure, how to name four-sided quadrilaterals, and how diagonals and regular vs irregular properties are determined.

Derive the angle sum property for quadrilaterals and polygons, showing the quadrilateral sum is 360 degrees and the general polygon sum is (n-2)×180.

practice problems on polygon interior angles, using the formula (n-2)180 to find n and sums for pentagon, quadrilateral, and hexagon, with examples.

Sum exterior angles of a polygon to 360 degrees, illustrated by a hexagon where each exterior angle is 60 degrees and interior plus exterior angles form a 180-degree linear pair.

Examine polygon angle sums: exterior angles total 360 degrees and interior sums equal (n-2)×180. Determine regular polygon feasibility by divisibility of 360, and solve a 1:5 ratio to find n=12.

Exterior theorem shows one equals three plus four; two equals five plus six. Hence one plus two equals A plus C.

compute the interior angle sum of a five-sided polygon using (n-2)180, then apply linear pairs to solve for x, which equals 80 degrees.

Solve for the polygon with nine diagonals, identify it as a hexagon, then find the interior angle as 120 degrees and the exterior angle as 60 degrees.

Split a general quadrilateral along a diagonal to form two triangles and sum their areas. With diagonal 7 cm and altitudes 2 cm and 3 cm, area = 17.5 cm².

Apply the diagonal-based area formula for a quadrilateral using heights 1 cm and 1.5 cm. Determine the diagonal length from area 20 cm², arriving at 16 cm.

Practice problem shows that doubling the diagonal and halving each height keeps the quadrilateral's area the same.

Explore parallelograms and their properties: opposite sides are equal and parallel; opposite angles are equal; diagonals bisect each other; adjacent angles are supplementary; midpoints of any quadrilateral form a parallelogram.

Explore why a parallelogram's opposite sides are parallel and equal, and why triangles ABC and DCB are congruent. Conclude that opposite angles are equal and diagonals bisect each other.

Practice problems apply angle relationships in parallelograms and triangles, using ratio reasoning, supplementary angles, and the 60/120 degree pattern.

Use angle chasing in a parallelogram configuration with base DC to deduce angle at C equals 60 degrees, via triangle BDC with 90 and 30 degree angles.

solve a parallelogram geometry problem by using opposite sides equal and parallel lines. determine angles B=60°, S=120°, R=60°, and apply diagonals bisecting to find PR=12 cm.

Solve a geometry problem using parallel lines and a transversal to find angle X. Apply angle sums, alternate interior angles, and supplementary relationships to determine X as 116 degrees.

Learn to find the area of a parallelogram using base and height by dropping a perpendicular to form a rectangle. See that equal perimeters don't guarantee equal areas.

Practice problem 1 shows how to use base times height to find the altitude for a side, extending a side and using distance between parallel lines to get 16/3 cm.

Doubling the side lengths of a parallelogram doubles its perimeter, showing that the new perimeter equals two times the original, and confirms option B as the correct choice.

Practice problem 3 applies right triangle geometry to find base, height, and missing side using area and Pythagoras, then compute triangle perimeter and area.

Define a rectangle as a parallelogram with opposite sides equal and right angles, then apply perimeter 2(l+b) and area l×b to 6×2 rectangle and 55×40 park with 1.5 m border.

Explore rectangle properties, show diagonals are equal and bisect each other in a parallelogram, and prove triangles ADC and BCT are congruent, yielding diagonals AC and BD.

Determine x by equating diagonal segments in a rectangle, since diagonals bisect and are equal. Set DG = 4x+19 and GB = 5x+1; 5x+1 = 4x+19 gives x = 18.

Compute the diagonal of a 10 cm by 24 cm rectangle using the Pythagoras theorem, showing that the diagonal equals 26 cm.

Solve a rectangle abcde angle problem by chasing angles, using vertical opposite angles and diagonals that are equal and bisect, deducing dca = 30 degrees and acb = 60 degrees.

Discover how to maximize the perimeter of a rectangle with constant area using integer sides, and how to maximize area with a perimeter by keeping dimensions as close as possible.

Maximize the rectangle's perimeter with integer sides by testing factor pairs of the fixed area, such as 1 and 24, yielding a maximum perimeter of 50 centimeters.

Given product 96, maximize X+Y with extreme factor pairs and minimize with closest pairs; given sum 24, maximize XY at 12 and 12, minimize at 1 and 23.

Explore a rectangle property: for any interior point, the sum of the squares of its distances to opposite vertices are equal, and apply it to find BC, yielding 4√2 cm.

Explore the square as a special rectangle with equal sides and 90-degree angles, and learn that diagonals are equal and perpendicular and bisect each other, defining its unique properties.

Learn why a quadrilateral can have at most three obtuse angles, since obtuse means over 90 degrees and the angle sum is 360 degrees, as shown by a rectangle.

Conclude that square diagonals intersect at 90 degrees to identify angle BQ, and that constructing a parallelogram requires at least three measurements: two non-parallel sides and one angle.

Convert the given rectangle areas from cm^2 to m^2 to work in one unit. Deduce the square’s side from integer-length constraints, and find the area is 144 m^2.

Explore quadrilaterals called trapezium, where one pair of opposite sides are parallel. Identify isosceles trapezium when the two parallel sides have equal length.

Identify trapezium when one pair of sides is parallel and interior angles are supplementary; recognize parallelograms where diagonals bisect each other, and note that rectangles and squares have 90-degree angles.

Solve this practice problem by recognizing parallel lines Abe and KDDI with a transversal and applying supplementary angles to obtain 125 degrees.

Apply parallel line properties and supplementary angles in practice problem 3 of ACT Math Prep to compute angle h e r as 130 degrees and another angle as 20 degrees.

Use parallel lines to establish 120° and 60° angles, show a portion forms an equilateral triangle, and apply parallelogram properties to deduce opposite sides equal; conclude X = 30 cm.

learn to find the area of a trapezium by using the height and the sum of its parallel sides, with area equal to half the height times that sum.

Drop perpendiculars to find the height of an isosceles trapezoid with bases 40 cm and 20 cm and legs 26 cm, height 24 cm, then compute the area.

Compute the area of an isosceles trapezoid with height 8 cm and legs 10 cm. Use the perimeter to find base sum 32 cm, yielding an area of 128 cm^2.

Identify consecutive equal sides and equal angles in a kite, then use diagonals: they intersect at 90 degrees, the longer diagonal bisects the shorter diagonal and is the angle bisector.

Analyze a quadrilateral with angles in ratio 3:7:6:4; rule out parallelogram, rhombus, and kite, and conclude it is a trapezium with one pair of parallel sides.

Explore kite geometry and right-angle diagonals, then solve a perimeter problem: with a 106-meter kite and one side of 23 meters, deduce the other sides as 23, 30, and 30.

Apply isosceles triangle properties to establish equal angles and use the 70 + 80 + 2x = 360 angle-sum to find x equals 105 degrees.

Solve a kite geometry problem to find x and y; deduce y equals 110 degrees, then use the sum of interior angles 360 to get x equals 80 degrees.

Practice a circle and tangent geometry problem by using exterior tangents, radii, and a kite, deriving angle relationships to conclude angle B is 75 degrees.

Practice circle geometry with tangents, radii, and angle chasing to show x or y equals 70 degrees in a rectangle with tangent lines and kite configurations.

Identify the isosceles triangle ABC within a circle of radius five centimeters; use radii and right triangles to show triangle OBC is equilateral, then conclude AC = 5 cm.

Explore rhombus properties, including equal sides, opposite angles, and diagonals that bisect each other at right angles, and how a rhombus differs from a square and a kite.

Construct the diagonals of rhombus ABCDE and use the given side-equals-diagonal condition. Recognize two equilateral triangles inside, yielding rhombus angles of 60 and 120 degrees.

Explore a rhombus diagonal property to find angle measurements: angle one equals 50 degrees, X equals 50, and Y minus X equals 40 degrees.

Apply the Pythagorean theorem to find x = 13 in the rhombus, then note diagonals bisect each other at 90 degrees to yield z = 13 and the perimeter 52.

Explore a rhombus geometry problem by proving congruent triangles through a perpendicular bisector, show EBD forms an equilateral triangle, and determine the rhombus angles as 60 and 120 degrees.

Learn to find the area of a rhombus by using its diagonals, which bisect at right angles, with area equal to one-half the product of the diagonals.

Learn to find the area of any polygon by splitting it into triangles, using altitudes, and summing half the base times the height for each triangle.

Compute the area of a rhombus using diagonals that bisect at right angles; with legs 3 cm and 4 cm, the rhombus area is 24 cm².

Determine the area of a rhombus with side 5 cm and diagonal 8 cm. Apply Pythagoras to get half-diagonals 4 cm and 3 cm, yielding 24 cm².

practice problem demonstrates that doubling both diagonals of a rhombus quadruples its area, using the formula area = (d1 × d2)/2.

Explore graphical division, a quick method that splits a figure into equal areas to locate the shaded region. See the half-area shortcut on a square of area 20 cm^2.

Divide a regular hexagon into six equilateral triangles to reveal six congruent parts, then reconfigure into twelve congruent right angled triangles to compute the hexagon's area as one twelfth.

Divide the regular hexagon (perimeter 18 cm) into six equilateral triangles to find a side length of 3 cm, then compute the area as 27 sqrt(3)/2 cm².

Explore a practice problem on a regular hexagon by dividing it into 12 equal parts to determine the shaded to unshaded area ratio as 1 to 5.

Explore isosceles trapezoids, where the non-parallel side equals the smaller base, and learn to partition trapezoids into equal-area parts. Analyze top-section divisions and six-part splits to see area relationships.

Solve a geometry problem in an isosceles trapezium with a 90-degree angle, using midpoints and perpendiculars to divide into six equal areas and show the shaded to unshaded ratio 1:5.

Apply graphical division to partition equilateral triangles and squares into equal-area parts, using midpoints to form four congruent triangles and exploring square divisions: two, four, eight, twelve, twenty four parts.

Determine the ratio of areas between triangles EDC and FGB in a parallelogram division, using congruent triangles and equal-area arguments.

Apply graphical division in a 20 cm square to quickly compute area ratios, yielding triangle areas 150 cm² and 100 cm² alongside the square’s 400 cm².

Compute the shaded area of a regular hexagon inscribed in a 20 cm diameter circle by dividing it into 12 equal parts and selecting four parts, yielding 50√3 cm².

Analyze a rectangle containing a square, showing that the triangle area is 8 cm², the square area is 32 cm², and five equal regions sum to 160 cm² for PQRS.

This lecture analyzes repeatedly inscribing a circle in a square and a square in that circle, revealing geometric progressions with ratio 1/2 and sums 2a^2 (squares) and pi a^2/2 (circles).

Explore a circle-square geometry problem with inscribed squares, calculate shaded regions using area differences and halves, and verify the result as 9/2 square centimeters.

Explore how to inscribe a circle, a hexagon, and a triangle inside a square, derive side lengths, compute areas, and compare perimeters using a unified geometric approach.

Compare the lap times for square, circle, hexagon, and equilateral triangle tracks at equal speeds, showing time is proportional to distance and yielding the ratio 4:2π:6:3√3.

Explore inscribing a circle in a square and compare their areas using a four-part layout. Calculate the square area as 16 cm², the circle area, and identify the shaded region.

In ACT Math Prep geometry problem, explore triangle and inscribed square, using Heron's formula, semiperimeter, and similarity to determine the square's side length.

Cut a circle from a square, then inscribe the largest possible square inside that circle; the final square's area equals half the area of the original square.

Calculate shaded area in a 20 cm square with four corner quarter circles by subtracting the circle area from the square; both cases give 86 cm² with pi ≈ 3.14.

Compute the remaining area by removing the largest square from a circle of radius 42 cm. The square’s diagonal is 84 cm, so the side is 84/√2, giving 2016 cm^2.

Compute the shaded area by subtracting the combined area of two semicircles (equivalently one circle of radius 14 cm) from the square of side 28 cm, yielding 168 square centimeters.

Explore the basics of circles, including circle definition, center, radius, diameter, chord, arc (minor and major), sector, segment, and the circumference, plus tangents and inscribed angles.

Learn circle geometry by defining circumference as the circle's border and applying c = 2 pi r and a = pi r^2, with pi approximations 22/7 or 3.14.

Set the rectangle area 14×11 equal to the circle area πr², substitute π = 22/7, and solve to find r = 7 cm.

Compute the distance in meters traveled by a wheel of radius 25 cm after 350 rotations using the circumference 2πr.

Learn circle properties including how a radius perpendicular to a chord bisects it, how equal chords are equidistant from the center, and the converse relationships.

Explore why the two tangents from an exterior point to a circle have equal lengths, using congruent right triangles formed by radii to the tangency points.

Solve a circumscribed quadrilateral problem using tangent segment equalities from exterior points. With AB = 8 cm and AD + BC = 11 cm, determine KD = 3 cm.

Solve a triangle with an incircle by using tangent lengths from an exterior point to relate segments and the semiperimeter, yielding a 5 cm value.

Solve a geometry problem involving an inscribed circle in a triangle, tangents from an exterior point, and kite properties to find angle cupie, which equals 105 degrees.

Explore circle properties: angle APB equals half the arc AB, and arc AB's measure equals the central angle AOB; minor versus major arcs determine the subtended angle.

Explore circle properties: angles subtended by the same arc are equal, and equal angles imply equal chords; use SAS congruence to relate chords AB and PR and converse.

Learn how a diameter yields a right angle at epb, and that the angle subtended by an arc equals half its measure; angles subtended by the same arc are equal.

Solve a circle geometry problem by applying inscribed angles subtending the same arc, diameter properties yielding right angles, and angle chasing to determine a 100-degree angle.

Apply circle geometry and right-triangle reasoning: with bq as diameter, pq = 10 cm; use pr = qs = 6 cm to find x ≈ 3.6 cm.

Explore an incircle problem in triangle ABC, using tangent radii creating right angles and the external angle theorem to prove the target angle equals 60 degrees.

Apply Pythagoras to a kite with four and three centimeter sides to find AM equals five and AB equals four point eight centimeters.

Explore the cyclic quadrilateral property, proving that opposite angles are supplementary by relating angles to arcs and the circle’s center.

Compute angle B using 360 minus the three given angles, identify the cyclic quadrilateral, apply opposite angles sum to 180, and determine RFQ is 50 degrees.

Use isosceles triangle properties to set equal angles and identify alternate interior angles. Apply cyclic quadrilateral reasoning and supplementary opposite angles to find the angle is 100 degrees.

Solve practise problem 3 by analyzing a cyclic quadrilateral, identify 90 and 45 degree angles, and apply isosceles and Pythagoras's theorem to find x = 3 and tr = 3√2.

Explore the secant tangent theorem, showing how the angle between a tangent and a secant equals the inscribed angle subtended by the same arc, with central and inscribed angle relationships.

Use tangent-angle relationships and triangle sums in a circle-tangent problem to determine the target angle, concluding it equals 80 degrees.

Use the angle bisector and second tangent theorem to set PSR = PSG = 50° and DSB = BRS = 50°, then find APS = 80° and PSU = 25°.

Use tangent–chord relations with a circle and tangents A and B; given angle ACB equals 60 degrees, conclude angle EDB also equals 60 degrees.

Learn the property for intersecting secants inside, outside, and tangent cases, using the point of intersection to relate segment products and solve problems.

analyze a semicircle problem by drawing a perpendicular from a point on the diameter to the circle and using intersecting chords to deduce 20 cm diameter, then compute its area.

Learn how to find cuboid volume as base area times height, compare to cylinders, and compute lateral, total surface areas, plus body diagonal length sqrt(L^2+B^2+H^2).

Explore the cube by defining side length L, compute volume as L^3, and determine total surface area as six times the surface area of one face across six faces.

Compute the surface area of an open-top box with dimensions 20 by 16 by 14 cm, yielding 1328 cm² per box, then multiply by 10 for 13280 cm².

Determine the water height in a rectangular field by applying volume equals base area times height, using 160 cubic meters and 800 square meters to get height 0.2 meters.

Find the maximum rod length in a 12 by 4 by 3 m room using the body diagonal. Compute sqrt(12^2+4^2+3^2)=sqrt(169)=13 meters.

Practice problem demonstrates counting painted faces on a sliced cube, computing total small cubes and identifying those with exactly one face painted, yielding 54 cubes.

Compute the surface area of the original four-centimeter cube and the total surface area of the 64 one-centimeter cubes, yielding a ratio of 1 to 4.

Compute how the cube's volume scales when its side multiplies by ten, and note that the volume becomes a thousand times the original.

This practised problem shows that a volume ratio of 1:64 gives side lengths in 1:4, so the area ratio of faces is 1:16 (the same as for total surface area).

Relate volume and capacity by showing a vessel's volume equals its capacity and master unit conversions among m^3, L, cm^3, and mL, including 1 m^3 = 1000 L.

Master right circular cylinder concepts, including lateral and total surface area and volume, and hollow cylinders with volumes pi (R^2 − r^2) h.

When the radius of a cylinder triples while the surface area remains unchanged, the height becomes one third of the original height.

Practice problem 2 shows that doubling the radius and reducing the height to one fourth keeps cylinder volume the same, since pi r^2 h equals pi (2r)^2 (h/4).

Apply cylinder volume rules to a two-cylinder ratio problem: with radii in 1:2 and heights in 2:3, derive a volume ratio of 1:6.

Roll a square sheet into a cylinder; set 2πr equal to the sheet’s side, so r = s/(2π) and the radius-to-side ratio is 1:2π.

Use a cylinder, 3:2 radius-height ratio, volume 19404 cm³; apply V = π r^2 h with π ≈ 22/7 to obtain r = 21 cm, h = 14 cm.

Calculate the hollow cylinder’s volume using outer radius 14 cm and thickness 2 cm (inner radius 12 cm), giving 11440 cm^3 and weight 91520 g at 8 g/cm^3.

Rotate a rectangle about its longer side to form a cylinder. Calculate the volume as 3850 cm^3 and the surface area as 1408 cm^2.

Explore right circular cone: height equals base radius r, slant height l; curved area pi r l, total area pi r l + pi r^2, volume one-third of cylinder.

Analyze a conical vessel with radius 9 cm and height 12 cm, where a spear is just immersed, to find the water spill fraction as 3/8.

Master sphere, hemisphere, and spherical shell concepts by using volume and surface area formulas: sphere 4/3 π r^3, hemisphere 2/3 π r^3, and shell 4/3 π(R^3−r^3) with area 4πR^2+4πr^2.

Equate the sphere and cone volumes to find the cone height, using 4/3 pi r^3 = 1/3 pi (2r)^2 h, giving h = r.

Solve a practice sphere problem: relate surface area to radius, derive rb = 3ra, then compare volumes to show a 96% decrease from B to A.

Compute the metal volume difference of a hollow shell with outer 20 cm and inner 10 cm diameters, using radii 10 cm and 5 cm and the (4/3)π(r^3) formula.

Explore faces, edges, and vertices of cubes, cuboids, and pyramids and prisms. Learn Euler's formula F + V = E + 2 and how it applies to these polyhedra.

Apply faces, vertices, and edges relation to solve questions. With 12 faces and 30 edges, deduce 20 vertices; the check shows eight faces and twelve edges cannot satisfy same condition.