Geometry Practice Test: Special Right Triangles (45-45-90 and 30-60-90)

In a 45-45-90 triangle the hypotenuse is leg · 2, so each leg is 10/2 = 52. The choice 102 multiplies by 2 instead of dividing.

The longer leg is the shorter leg times 3: 63. The value 12 is the hypotenuse (twice the shorter leg).

Shorter leg = longer leg ÷ 3 = 9/3 = 93/3 = 33. The value 4.5 halves the longer leg, but halving works only from the hypotenuse.

A diagonal splits the square into two 45-45-90 triangles, so the side is 8/2 = 42 cm and the area is (42)² = 32 cm². The area 64 cm² treats the diagonal as a side.

The height splits the triangle into two 30-60-90 triangles. The height is the longer leg, so the shorter leg is 6 and the side (the hypotenuse) is 12. The perimeter is 3 · 12 = 36.

The ramp is the hypotenuse and the rise is opposite the 30° angle, so it is the shorter leg: 14/2 = 7 ft. The value 73 ft is the horizontal distance.

The diagonal and two sides form a 30-60-90 triangle with hypotenuse 20, so the shorter side is 10 and the longer side is 103. The area is 10 · 103 = 1003.

A regular hexagon is made of six equilateral triangles with side 4, and the apothem is the height of one of them. Half the side is 2, so the height is 23.

The hypotenuse is 32 · 2 = 6, so the perimeter is 32 + 32 + 6 = 6 + 62. The choice 6 + 32 counts only one leg.

The ratio must be 1 : 3 : 2, and 5 : 53 : 10 fits. The set 5, 10, 55 is a right triangle but its legs are in a 1 : 2 ratio, and 5, 5, 52 is a 45-45-90 triangle.