Geometry Practice Test: Right Triangles and Trigonometry

By the Pythagorean theorem, c² = 9² + 12² = 81 + 144 = 225, so c = 15. The value 225 is c², and 21 is just 9 + 12.

The missing leg satisfies b² = 26² − 10² = 676 − 100 = 576, so b = 24. The value 16 is just 26 − 10, but side lengths of a right triangle combine through their squares.

Only 7² + 24² = 49 + 576 = 625 = 25². For 5, 11, 12 the check gives 146, not 144, so that triangle is not right.

Compare the square of the longest side with the sum of the other two squares: 10² = 100 is greater than 6² + 7² = 85, so the angle opposite the side of 10 is obtuse. The triangle does exist, since 6 + 7 > 10.

In a 45°-45°-90° triangle the hypotenuse is the leg times 2, so it is 72 ≈ 9.9. The factor 3 belongs to the long leg of a 30°-60°-90° triangle.

The short leg (opposite 30°) is half the hypotenuse, 9, and the long leg (opposite 60°) is the short leg times 3: 93. The value 9 is the leg opposite the 30° angle.

Sine is opposite over hypotenuse. The side opposite angle A is BC = 8 and the hypotenuse is 17, so sin A = 8/17. The ratio 15/17 is cos A, and 8/15 is tan A.

The height is opposite the 35° angle and the 50 m distance is adjacent, so h = 50 tan 35° ≈ 35.0 m. Using sine gives 28.7 m and dividing by the tangent gives 71.4 m, both of which use the wrong ratio.

tan θ = 5/12, so θ = tan⁻¹(5/12) ≈ 22.6°. The value 24.6° comes from using sin⁻¹(5/12), but 12 is a leg, not the hypotenuse; 67.4° is the other acute angle.

The sine of an angle equals the cosine of its complement, so x = 90 − 32 = 58. The values 122 and 148 are outside the allowed range.

The ladder is the hypotenuse: h² = 13² − 5² = 169 − 25 = 144, so h = 12 feet. The value 13.9 feet comes from adding the squares, which treats the ladder as a leg.

cos 40° = adjacent/hypotenuse, so the hypotenuse is 10 ÷ cos 40° ≈ 13.05 cm. Multiplying instead (10 cos 40° ≈ 7.66) gives a length shorter than the leg, which is impossible for a hypotenuse.

The altitude splits the triangle into two 30°-60°-90° triangles with hypotenuse 10 and short leg 5. The altitude is the long leg: 53 ≈ 8.66.

The angle of depression equals the angle of elevation from the boat, so tan 12° = 60/d and d = 60 ÷ tan 12° ≈ 282.3 m. The value 288.6 m is the straight-line distance to the top (60 ÷ sin 12°), not the horizontal distance.

The altitude to the hypotenuse is the geometric mean of the two segments: 4 · 9 = 36 = 6. The value 6.5 is the arithmetic mean, and 36 is the product before taking the square root.