Pythagoras and right-angled trigonometry

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

1. AC
2. AB
3. BC
4. Angle B

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

1. 18 cm
2. 15 cm
3. 21 cm
4. √63 cm

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

1. 9 m
2. 19 m
3. 15 m
4. √353 m

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

1. √12 cm
2. 6 cm
3. 12 cm
4. √74 cm

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?

1. 8 m
2. 4 m
3. 11.7 m
4. 16 m

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

1. 8 units
2. 10 units
3. 12 units
4. 14 units

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

1. It is right-angled because 7 + 24 = 25.
2. It is not right-angled because 25 is the longest side.
3. It is right-angled because 7² + 24² = 25².
4. It is equilateral because all three lengths are positive.

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

1. AC/AB
2. BC/AC
3. AB/BC
4. 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?

1. 7 cm
2. 12.1 cm
3. 14 cm
4. 28 cm

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?

1. 6 m
2. 9 m
3. 15.6 m
4. 36 m

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.

1. 7.7 cm
2. 14.3 cm
3. 10.1 cm
4. 18.6 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.

1. 17.0°
2. 73.7°
3. 86.6°
4. 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.

1. 61.9°
2. 28.1°
3. 43.8°
4. 62.8°

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

1. Cosine
2. Sine
3. Tangent
4. Pythagoras only

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.

1. 6.3 m
2. 9.0 m
3. 13.4 m
4. 19.2 m

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

1. 5 units
2. 12 units
3. √74 units
4. 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?

1. 5 units
2. 1 unit
3. 7 units
4. 9 units

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.

1. 23.9 m
2. 28.1 m
3. 38.2 m
4. 72.0 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.

1. 23.4 m
2. 37.0 m
3. 25.0 m
4. 39.1 m

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.

1. 20.4 m
2. 35.6 m
3. 87.2 m
4. 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?

1. 15 m
2. 12.7 m
3. 18 m
4. 21 m

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.

1. 12.0°
2. 11.3°
3. 78.7°
4. 80.0°

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.

1. 36.9°
2. 46.7°
3. 53.1°
4. 66.2°

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?

1. It is measured along any segment from P that meets l.
2. It is always measured horizontally from P.
3. It is always the longest segment from P to l.
4. It is measured along the segment from P perpendicular to l.