SAT and ACT Math - Basic Geometry concepts

Compute slope from two points using m = (y2 - y1) / (x2 - x1) and relate this to slope intercept form y = mx + c.

Identify the intersection by equating the lines' y-values. Demonstrate with y = x + 2 and y = 2x + 1 to solve x = 1 and y = 3.

Find the equation of a line through two points using the two-point form. For points (2,3) and (4,6), this gives y = (3/2)x with zero y-intercept.

Identify parallel lines by equal slopes, then derive the line through (3,4) parallel to y = 2x + 3, using y = 2x + c and c = -2.

find the equation of a line perpendicular to a given line by using the negative reciprocal of its slope and a point; for y=2x+1 through (2,3) it is y=-1/2 x+4.

Learn to find x and y intercepts of a line by substitution: x=0 gives y-intercept, y=0 gives x-intercept; vertical lines have undefined slope, horizontal lines have slope zero.

Calculate the distance between two points on a plane using the distance formula, shown with (2,3) and (4,5), yielding the distance square root of eight units.

Compute the midpoint of two points on a line using the midpoint formula. Average the x-coordinates and the y-coordinates to obtain (3.5, 3.5) as the midpoint.

Calculate the circumference of a circle using c = 2πr with r = 7 cm; applying π = 22/7 yields a circumference of 44 cm.

Using area formula a = pi r^2 with pi approximated as 22/7 and r = 7 cm, the area becomes (22/7) * 7^2, yielding 164 square centimeters.

Compute arc length using the central angle and radius with the formula length = (theta/360) × 2πr, shown for a 7 cm circle and 18° angle, giving about 2.2 cm.

Calculate the length of the chord in a circle using B = 2 r sin(theta/2). With r = 10 cm and theta = 60°, the chord length is 10 cm.

Find the distance from the center to a chord by drawing the radius to the chord, dropping a perpendicular to bisect it, and applying Pythagoras.

Compute the area of a sector from a 40-degree angle, using the radius squared, and conclude the sector area is eight thousand square centimeters.

The example shows finding the area of a circle segment by subtracting triangle area from the sector, using a 10 cm radius and 16 cm chord, yielding 52 square centimeters.

derive the circle equation in standard form (x-h)^2+(y-k)^2=r^2, identify center (h,k) and radius r, and cover special cases: origin, x-axis, and y-axis centers with example (8,3) and r=7.

Define the perimeter of a triangle as the sum of its three side lengths; for sides 6, 7, and 9, the perimeter is 22 units (22 centimeters).

Compute the area of a triangle from three sides by using the semiperimeter s = (a+b+c)/2 and the area sqrt[s(s-a)(s-b)(s-c)], illustrated with sides 6, 7, 9.

Compute triangle area from base and height using the formula area equals one-half base times height, demonstrated with base five cm and height four cm.

Identify equilateral triangles where all sides are equal, apply the area formula sqrt(3)/4 times side squared, and compute for a 6 cm side to get 9 sqrt(3) square centimeters.

Learn the height of an equilateral triangle by dropping a perpendicular to the base, splitting it in half, and applying Pythagoras.

Define a median as a line from a vertex to the midpoint of the opposite side. Converge at the centroid, where each is divided in a 2:1 ratio.

Identify the incenter by the intersection of the triangle's angle bisectors, the center of a circle tangent to all three sides.

Learn how to find the circumcenter of a triangle, a point equidistant from all three vertices, by drawing the perpendiculars from the vertices to the opposite sides.

Explore right angled triangles, apply c^2 = a^2 + b^2 to relate hypotenuse and legs, and practice calculating the hypotenuse from two given sides, as in a 4–3–5 example.

Explore Pythagorean triplets: sets of positive integers a, b, c that satisfy a^2 + b^2 = c^2 with c as the hypotenuse. See the example 5, 12, 13.

Explore trigonometric functions in right triangles, defining sine, cosine, and tangent from opposite, adjacent, and hypotenuse, and use reciprocals cosecant, secant, and cotangent in a 3-4-5 triangle example.

Explore rhombus properties in sat and act geometry, including all four sides equal, opposite sides parallel, and diagonals that bisect each other and are perpendicular.

Split a rhombus into two triangles with a diagonal to apply base times height. For diagonals 4 and 6, area equals 12 square centimeters.

In a rhombus, diagonals satisfy p^2 + q^2 = 4 s^2. Given a side of 4 cm and one diagonal of 3 cm, the other diagonal is sqrt(55) cm.

Master parallelogram properties, including opposite sides being equal and opposite angles equal when lines intersect, within basic geometry concepts for SAT and ACT math.

Learn to compute a parallelogram's area by decomposing it into triangles and rectangles, using base times height and half base times height, then sum the parts.

Use the diagonal relation d1^2 + d2^2 = 2(a^2 + b^2) to find diagonal for sides 2 and 4 when a diagonal is 5 cm; the other is sqrt(15) cm.

Explore trapezoids, a four-sided shape with one pair of parallel sides, including regular and irregular types, where the parallel sides may differ in length and the other sides vary.

Learn to compute trapezoid area using the formula A = (h/2)(a + b), with bases a and b and height h, illustrated with 4 cm and 6 cm bases.

Explain that a square has four equal sides and two diagonals intersecting at right angles, then use Pythagoras to find the diagonal and compute area from the side.

Learn rectangle properties: opposite sides equal and four right angles. Calculate perimeter, area, and diagonal using formulas: perimeter 2l+2w, area l×w, diagonal sqrt(l^2+w^2).

Explore quadrilaterals, four-sided shapes such as square, rectangle, rhombus, and trapezoid, and distinguish convex from concave by diagonals; interior angles always sum to 360 degrees.

Discover the key property of cyclic quadrilaterals: opposite angles sum to 180 degrees, and all four vertices lie on a common circle.

Discover how to find the number of diagonals in a polygon using the n choose 2 minus n formula, with pentagon and octagon examples.

Discover cube geometry by recognizing its six square faces, compute surface area as six times side squared, and determine volume as side cubed using centimetres.

Discover that a cuboid has six faces with opposite faces equal, then calculate its surface area from length, width, and height and its volume from those dimensions.

Explore sphere geometry by linking surface area and volume to the radius, using area 4 pi r^2 and volume 4/3 pi r^3, r = 7 cm, pi = 22/7.

Explore hemisphere formulae for surface area and volume, including surface area as 3 pi r^2, and volume as half of a sphere.

Master cylinder formulae by calculating surface area and volume using radius and height. Apply the formulas 2πr(h+r) for total surface area and πr^2h for volume, with a practical example.