Geometry for Secondary or High School [Full Course]

Relate the distances of three collinear points P, Q, and M by showing PQ plus QM equals PM, establishing the distance relation on a line.

Apply triangle inequality to bound distance from City A to City C: it lies between 21 km and 551 km, given City B is 265 km northwest of City A.

Solve the distance problem from practice exercise four: with XY = 5 units and XZ = 1.5 units, determine ZY as 3.5 units.

This lecture demonstrates finding X in angle problems using straight-line and rotation, computing X as 180 minus 40, 360 minus 68, and 146 minus 35 yield 140, 292, 111 degrees.

Explains proof as a sequence of statements and uses lines K and L to show opposite angles have equal measures through 180-degree straight angles.

Demonstrates how perpendicular lines form right angles and proves that angle A equals angle B by using equal 90-degree relationships.

Solve practice exercise 4 on angles of a triangle by applying supplementary angles, parallel lines with a transversal, and angle sums in triangles to find x in multiple figures.

Use angle-bisector reasoning in triangle ABC with angle C = 70°, set A = 2x and B = 2y, get x+y = 55°, then find angle P = 125°.

Practice exercise 1 teaches plotting points on a coordinate system by labeling the x-axis and y-axis, locating the origin, and marking points such as (1,-1), (0,5), (-5,-2), and (1,0).

Explore coordinate systems by inspecting quadrants: in the third quadrant, x is negative and y is negative, with the origin and axes guiding sign identification.

Plot the points (1.5, -1) and (-1.5, -1) on a coordinate system, identify the origin, and describe positive and negative directions on the axes.

Learn to find line's equation from a given slope and point using y - y1 = m(x - x1). Practice slope 4 through (1,1) and slope -1/2 through (sqrt(2), 3).

Determine the square's side that matches the rectangle's area of 400, showing a square with side 20.

Compute the area of the one-meter-wide path around a 15 by 20 meter garden by subtracting the garden from the enclosing 17 by 22 meter rectangle, yielding 74 square meters.

Compare right triangles ABC and XYZ whose legs double; use the area formula 1/2 base times height to find ABC's area 1/2 ab and XYZ's 2ab, yielding the 1:2 ratio.

Calculate the total area of the track by summing areas of three attached trapezoids using the formula area = 1/2(base1+base2)×height, with height 25 meters and bases 40/30, 55/40, and 25/55.

Apply Pythagoras theorem to a cube with equal side lengths to derive its space diagonal. When the side is 2, the diagonal is 2 sqrt(3).

Apply the Pythagoras theorem to a right triangle with legs 25 m and 30 m to find the wire length; it equals sqrt(1525) = 5 sqrt(61) meters, about 39 meters.

Apply the Pythagoras theorem to find the field diagonal from 18 meters and 30 meters, then compute the wire length with 20 meters and that diagonal, yielding about 40 meters.

Analyze practice exercise 13 as it applies the Pythagoras theorem to find PQ's length via a line ST parallel to PQ, yielding PQ = 8 sqrt(6).

Calculate the side length of an equilateral triangle with a 12-meter perimeter, then use the height from the Pythagorean theorem to compute the area as 2 sqrt(12) square meters.

Use the right angle at MOP and isosceles relations MP = PQ to determine MP = sqrt(2). Then apply the Pythagorean theorem to find MQ as sqrt(4 + 2√2).

Apply the distance formula to find the distance between points P and Q in two examples, yielding sqrt(106) in both cases.

Apply the distance formula to find the y-axis point equidistant from (0,-32) and (0,51); solve a^2 = b^2 to obtain y = -13/2, giving the point (0,-13/2).

Plot the square's vertices at (4,0), (4,4), (8,4), and (8,0) to visualize the shape, then compute the side length as 4 and find the area as 16.

Apply the distance formula in three-dimensional space to compute distances between points with triple coordinates, yielding d = sqrt(13) for A and d = 5 sqrt(2) for B.

Determine the circle centered at (3,5) with radius 5 using the distance formula, then find the other y-axis intersection (0,9) from the standard equation.

Explore circle theorems: chords, diameters, and tangents. Learn that an inscribed angle equals half its intercepted arc, with three cases, and that a diameter subtends a right angle.

Practice exercise 1 on theorems about circles guides you to find X in diagrams using half-angle reasoning. It includes cases like X equals half of 360 minus 110.

In concentric circles with radii 3 and 7, the tangent chord AB has length 4 sqrt(10), derived from a right triangle with legs 3 and TB and hypotenuse 7.

Explore polygons by identifying sides and vertices, distinguishing regular and irregular polygons, and convex and concave shapes, and applying the interior angle sum formula (n-2)×180 degrees.

Apply the formula (s - 2) × 180 ÷ s to find each angle of a regular polygon for A, B, and C with 6, 11, and 14 sides.