Triangles

The method

1. Identify the hypotenuse opposite the right angle.
2. Use c² = a² + b² when two sides are known. Subtract squares to find a shorter side.
3. Relative to the chosen angle, label the opposite, adjacent and hypotenuse sides.
4. Use sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse or tan θ = opposite ÷ adjacent.
5. For a compound problem, solve the first triangle and carry the unrounded shared length into the second.

Mistakes that lose marks

• Adding squares when the hypotenuse is already known.
• Using an angle from a different triangle.
• Rounding a shared length before finishing.
• Accepting a shorter side longer than the hypotenuse.

EXAM-STYLE questions written by Apex tutors

Practise with 9 original questions and full solutions. The suggested marks are a guide from our tutors. M means method, A means accuracy and B means an independent point. Try each question, then open its solution.

1. Apex original core A right triangle has shorter sides 20 m and 21 m. Find the hypotenuse. [2 marks]
   Show full solution

   1. x² = 20² + 21² = 400 + 441 = 841. M1
   2. x = √841 = 29 m. A1

   Answer: 29 m

2. Apex original extended A right triangle has hypotenuse 26 cm and one shorter side 10 cm. Find the other shorter side. [3 marks]
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   1. y² = 26² − 10² = 676 − 100 = 576. M1
   2. y = √576 = 24 cm. A2

   Answer: 24 cm

3. Apex original extended A right triangle has hypotenuse 18 cm and angle A = 27°. Find the side opposite A to 3 significant figures. [3 marks]
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   1. sin 27° = p ÷ 18. M1
   2. p = 18 × sin 27° = 8.171829… cm = 8.17 cm. A2

   Answer: 8.17 cm

4. Apex original extended A right triangle has side 12 m adjacent to angle B = 48°. Find the hypotenuse to 3 significant figures. [3 marks]
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   1. cos 48° = 12 ÷ q. M1
   2. q = 12 ÷ cos 48° = 17.933719… m = 17.9 m. A2

   Answer: 17.9 m

5. Apex original extended Relative to angle θ, a right triangle has opposite side 7 cm and adjacent side 15 cm. Find θ to 1 decimal place. [3 marks]
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   1. tan θ = 7 ÷ 15. M1
   2. θ = tan⁻¹(7 ÷ 15) = 25.016893…° = 25.0°. A2

   Answer: 25.0°

6. Apex original extended A ladder reaches 3.0 m up a vertical wall when its foot is 1.6 m from the wall. Find its length. The same ladder is then moved so its foot is 3.2 m from the wall. Find its new height to 3 significant figures. [5 marks]
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   1. Length squared = 3.0² + 1.6² = 9 + 2.56 = 11.56. M1
   2. Length = √11.56 = 3.4 m. M1
   3. New height squared = 3.4² − 3.2² = 11.56 − 10.24 = 1.32. M1
   4. New height = √1.32 = 1.148913… m = 1.15 m. A2

   Answer: Length 3.4 m; new height 1.15 m

7. Apex original extended A roof apex is 3.0 m vertically above a horizontal base. The horizontal distances from the point below the apex to the left and right ends are 4.8 m and 6.0 m. Find both sloping beam lengths to 3 significant figures and the left base angle to 1 decimal place. [6 marks]
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   1. Left length squared = 4.8² + 3.0² = 23.04 + 9 = 32.04. M1
   2. Left length = √32.04 = 5.660389… m = 5.66 m. M1
   3. Right length squared = 6.0² + 3.0² = 36 + 9 = 45. M1
   4. Right length = √45 = 6.708204… m = 6.71 m. M1
   5. Left angle = tan⁻¹(3.0 ÷ 4.8) = 32.005383…° = 32.0°. A2

   Answer: Left 5.66 m; right 6.71 m; angle 32.0°

8. Apex original extended A vertical tower stands on level ground. P and Q lie on the same straight line on one side of its base. Q is 12 m farther from the base than P. Elevation angles are 52° at P and 35° at Q. Let the height be h and the distance from P to the base be d. Form two equations and find h to 1 decimal place. [6 marks]
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   1. h = d tan 52° and h = (d + 12) tan 35°. M1
   2. d(tan 52° − tan 35°) = 12 tan 35°. M1
   3. d = (12 × tan 35°) ÷ (tan 52° − tan 35°) = 14.493697… m. M1
   4. Using the unrounded distance, h = d tan 52° = 18.551087… m = 18.6 m. A3

   Answer: 18.6 m

9. Apex original extended A rectangle has sides 8 cm and 6 cm. Find its diagonal and the angle between the diagonal and the 8 cm side to 1 decimal place. [5 marks]
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   1. Diagonal squared = 8² + 6² = 64 + 36 = 100. M1
   2. Diagonal = √100 = 10 cm. M1
   3. Angle = tan⁻¹(6 ÷ 8) = 36.869898…° = 36.9°. A3

   Answer: 10 cm; 36.9°

Where this is in the syllabus

Explore the matching courses: Cambridge O Level Mathematics (Syllabus D) 4024.

For the official syllabus, use the board links: Cambridge O Level Mathematics (Syllabus D) 4024.

Questions students ask