Pythagoras and right-angled trigonometry

1. Triangle ABC is right-angled at B. Which side is the hypotenuse?

Answer A: AC

The right angle is at B, so the side opposite B is AC. Therefore AC is the hypotenuse. AB and BC meet at the right angle and are the perpendicular legs.

2. A right-angled triangle has perpendicular legs of 9 cm and 12 cm. What is its hypotenuse?

Answer B: 15 cm

The hypotenuse c satisfies c² = 9² + 12² = 81 + 144 = 225. Hence c = √225 = 15 cm, which is longer than either leg.

3. A right-angled triangle has hypotenuse 17 m and one leg 8 m. What is the other leg?

Answer C: 15 m

The missing leg x satisfies x² = 17² - 8² = 289 - 64 = 225. Therefore x = 15 m. The check 8² + 15² = 17² confirms the answer.

4. A right-angled triangle has perpendicular legs of 5 cm and 7 cm. What is the exact length of its hypotenuse?

Answer D: √74 cm

The hypotenuse is √(5² + 7²) = √(25 + 49) = √74 cm. Since 74 has no square factor greater than 1, this exact surd cannot be simplified further.

5. A straight 10 m ladder rests against a vertical wall. Its foot is 6 m horizontally from the wall on level ground. How high up the wall does it reach?

Answer A: 8 m

If the vertical height is h, then h² + 6² = 10². Thus h² = 64 and h = 8 m. The positive root is used for the physical height.

6. Point P is (-2, 1) and point Q is (4, 9). What is the straight-line distance PQ?

Answer B: 10 units

The horizontal change is 4 - (-2) = 6 and the vertical change is 9 - 1 = 8. Therefore PQ = √(6² + 8²) = √100 = 10 units.

7. Which statement is correct for a triangle whose side lengths are 7 cm, 24 cm and 25 cm?

Answer C: It is right-angled because 7² + 24² = 25².

Since 7² + 24² = 49 + 576 = 625 and 25² = 625, the Pythagorean equality holds. The angle opposite the 25 cm side is therefore 90°.

8. Triangle ABC is right-angled at C. Relative to acute angle A, which ratio equals sin A?

Answer D: BC/AB

Because angle C is 90°, AB is the hypotenuse. Relative to angle A, BC is opposite. Therefore sin A = BC/AB.

9. In a right-angled triangle, an acute angle is 60° and the hypotenuse is 14 cm. What is the length of the leg adjacent to the 60° angle?

Answer A: 7 cm

The adjacent length is 14 cos 60° = 14 × 0.5 = 7 cm. It must be shorter than the 14 cm hypotenuse.

10. In a right-angled triangle, an acute angle is 30° and the hypotenuse is 18 m. What is the length of the leg opposite the 30° angle?

Answer B: 9 m

The opposite length is 18 sin 30° = 18 × 0.5 = 9 m. This is half the hypotenuse, as expected for the side opposite 30° in a right triangle.

11. A right-angled triangle has an acute angle of 40° and an adjacent leg of 12 cm. Find the opposite leg, correct to one decimal place.

Answer C: 10.1 cm

Let the opposite leg be x. Then tan 40° = x/12, so x = 12 tan 40° = 10.069.... Rounded to one decimal place, the length is 10.1 cm.

12. Triangle RST is right-angled at S. Relative to angle R, ST = 7 cm is opposite and RS = 24 cm is adjacent. Find angle R correct to one decimal place.

Answer D: 16.3°

tan R = 7/24, so R = tan⁻¹(7/24) = 16.260.... To one decimal place, angle R is 16.3°.

13. Triangle UVW is right-angled at V. The hypotenuse UW is 17 m and UV = 8 m is adjacent to angle U. Find angle U correct to one decimal place.

Answer A: 61.9°

cos U = 8/17, so U = cos⁻¹(8/17) = 61.927.... Therefore angle U is 61.9° to one decimal place.

14. Which trigonometric ratio should be used first when the opposite leg and hypotenuse are known and the acute angle is required?

Answer B: Sine

Sine relates opposite and hypotenuse: sin θ = opposite/hypotenuse. The angle is then found using inverse sine.

15. In a right-angled triangle, an acute angle is 35° and its adjacent leg is 11 m. Find the hypotenuse correct to one decimal place.

Answer C: 13.4 m

If the hypotenuse is h, then cos 35° = 11/h, so h = 11/cos 35° = 13.428.... Rounded to one decimal place, h = 13.4 m.

16. Point A has coordinates (5, -7). What is the shortest distance from A to the x-axis?

Answer D: 7 units

The x-axis is the horizontal line y = 0. The perpendicular vertical distance from y = -7 to y = 0 is |-7| = 7 units.

17. Point P is (-2, 4), and line l is the vertical line x = 3. What is the shortest distance from P to line l?

Answer A: 5 units

The horizontal distance from x = -2 to x = 3 is |3 - (-2)| = 5 units. This horizontal segment is perpendicular to the vertical line, so it is the shortest distance.

18. From point A on level ground, the foot B of a vertical tower is 45 m away horizontally. The angle of elevation from A to the top of the tower is 32°. Find the tower height correct to one decimal place.

Answer B: 28.1 m

If the height is h, tan 32° = h/45. Therefore h = 45 tan 32° = 28.119..., which rounds to 28.1 m.

19. An observer's eyes are 1.6 m above level ground. The observer is 30 m horizontally from a vertical tree and sees its top at an angle of elevation of 38°. Find the tree's total height correct to one decimal place.

Answer C: 25.0 m

The rise above eye level is 30 tan 38° = 23.4386... m. Adding 1.6 m gives 25.0386... m, so the total height is 25.0 m to one decimal place.

20. A point C is at the top of a vertical 40 m cliff above foot F. A boat B is at sea level, FB is horizontal, and the angle of depression from C to B is 27°. Find FB correct to one decimal place.

Answer D: 78.5 m

Using the equal 27° angle at B, tan 27° = 40/FB. Hence FB = 40/tan 27° = 78.504..., so the horizontal distance is 78.5 m.

21. A vertical pole is 9 m tall. A straight wire joins its top to a ground anchor 12 m horizontally from its foot. How long is the wire?

Answer A: 15 m

The wire length is √(9² + 12²) = √225 = 15 m. It is longer than either perpendicular leg and shorter than their 21 m sum.

22. A straight ramp rises 1.2 m while extending 6 m horizontally. Find its angle of elevation from the horizontal, correct to one decimal place.

Answer B: 11.3°

tan θ = 1.2/6 = 0.2, so θ = tan⁻¹(0.2) = 11.3099.... The angle of elevation is therefore 11.3° to one decimal place.

23. A vertical mast CT, anchor A and observer B lie in one vertical plane on level ground. Support cable AT is 25 m, AC is 7 m horizontally, and BC is 18 m horizontally. Find the angle of elevation from B to mast top T correct to one decimal place.

Answer C: 53.1°

Pythagoras gives CT = √(25² - 7²) = 24 m. Then tan θ = CT/BC = 24/18, so θ = tan⁻¹(24/18) = 53.1301..., or 53.1° to one decimal place.

24. Point P is not on straight line l, and both lie in the same plane. Which statement correctly describes the shortest distance from P to l?

Answer D: It is measured along the segment from P perpendicular to l.

The shortest distance from a point to a straight line is the length of the perpendicular segment joining them. It is horizontal only when the given line is vertical.