Pythagoras' Theorem

Ryan · G8 Mathematics
Based on Cambridge IGCSE Mathematics (0580) syllabus
Created by Miss Clarissa Ng
www.clartutors.com
Instructions: This worksheet has 3 sections. Answer all questions. Section A: Foundation (10 questions). Section B: Core Application (10 questions). Section C: Challenge (5 questions). Show all working. Use a calculator. Total: 25 questions.

Key Concepts Covered

Section A

Foundation

~25 min

Find the unknown length. Give your answers correct to 2 decimal places unless stated otherwise.

1
7 cm 24 cm c Find the length of the hypotenuse c.
2
12 cm b 37 cm Find the length of the shorter side b.
3
x 15 cm 17 cm Find the length of x.
4
9.6 cm y 20.3 cm Find the length of y.
5
5.3 cm 8.1 cm z Find the length of the hypotenuse z.
6
11.4 cm 6.7 cm w Find the length of the hypotenuse w.
7
a 4.2 cm 10.5 cm Find the length of a.
8
3.6 cm 7.8 cm d Find the length of the hypotenuse d.
9
e 13.7 cm 18.2 cm Find the length of e.
10
22.5 cm f 30.1 cm Find the length of f.
Section B

Core Application

~45 min

Show all your working. Give answers correct to 2 decimal places unless stated otherwise.

11
h 16 cm 17 cm 17 cm A C B Triangle ABC is isosceles with AB = BC = 17 cm and AC = 16 cm. Find the height h from B to AC.
12
12 cm 9 cm d A rectangle has sides 12 cm and 9 cm. Find the length of the diagonal d.
13
6 m h 10 m A ladder 10 m long leans against a vertical wall. The foot of the ladder is 6 m from the wall. How high up the wall does the ladder reach?
14
8 cm 10 cm 3 cm 3 cm B D A C ABCD is a kite. The diagonals intersect at right angles at O. AO = OC = 3 cm, BO = 8 cm and OD = 10 cm. Find the length of side BC.
15
8 cm 5 cm 3 cm D A cuboid has dimensions 8 cm × 5 cm × 3 cm. Find the length of the space diagonal D.
16
6 cm 6 cm 4 cm 4 cm Diagonals bisect at right angles A rhombus has diagonals of 12 cm and 8 cm. Find the length of one side of the rhombus.
17
h 20 cm 12 cm 10 cm An isosceles trapezium has parallel sides of 20 cm and 12 cm. The non-parallel sides are each 10 cm long. Find the height h of the trapezium.
18
A right-angled triangle has a hypotenuse of 25.6 cm and one side of 19.3 cm. Find the area of the triangle. Give your answer correct to 2 decimal places.
19
15 cm 12 cm 9 cm A B C Triangle ABC has AB = 15 cm, BC = 12 cm and AC = 9 cm. Determine whether ABC is a right-angled triangle. Show your working clearly.
20
Determine whether each of the following sets of sides forms a right-angled triangle. Show your working.
  • (a) Sides 7.2 cm, 9.6 cm, 12.0 cm
  • (b) Sides 14 cm, 48 cm, 50 cm
  • (c) Sides 5.5 cm, 7.3 cm, 9.1 cm
Section C

Challenge

~30 min

These questions require multiple steps. Show all working clearly.

21
pos 1 pos 2 5 m 8 m wall A 13 m ladder leans against a vertical wall. In position 1, the foot is 5 m from the wall. In position 2, the foot is 8 m from the wall.
  • (a) How high does the ladder reach in each position?
  • (b) By how many metres does the top of the ladder drop?
22
h 18 cm 18 cm 18 cm Triangle ABC is equilateral with each side 18 cm. Find (a) the height h and (b) the area of the triangle.
23
13 cm 15 cm 20 cm 16 cm P A B D C Rectangle ABCD has AB = 20 cm and BC = 16 cm. Point P is inside the rectangle. PA = 13 cm and PB = 15 cm. Find the distance from P to side AD.
24
A ship sails 18 km due north and then 24 km due east.
  • (a) How far is the ship from its starting point?
  • (b) A second ship sails 30 km north and 40 km east. Which ship is further from the starting point?
  • (c) If the first ship returns directly to the start at 12 km/h, how long does the return journey take?
25
24 cm 18 cm d A rectangular sheet of paper measures 24 cm by 18 cm. It is folded along the diagonal. Prove that the diagonal has length 30 cm, and find the area of the triangle formed by the fold.

Answer Key

Q1: √(7² + 24²) = √(49 + 576) = √625 = 25 cm
Q2: √(37² − 12²) = √(1369 − 144) = √1225 = 35 cm
Q3: √(17² − 15²) = √(289 − 225) = √64 = 8 cm
Q4: √(20.3² − 9.6²) = √(412.09 − 92.16) = √319.93 = 17.89 cm
Q5: √(5.3² + 8.1²) = √(28.09 + 65.61) = √93.7 = 9.68 cm
Q6: √(11.4² + 6.7²) = √(129.96 + 44.89) = √174.85 = 13.22 cm
Q7: √(10.5² − 4.2²) = √(110.25 − 17.64) = √92.61 = 9.62 cm
Q8: √(3.6² + 7.8²) = √(12.96 + 60.84) = √73.8 = 8.59 cm
Q9: √(18.2² − 13.7²) = √(331.24 − 187.69) = √143.55 = 11.98 cm
Q10: √(30.1² − 22.5²) = √(906.01 − 506.25) = √399.76 = 19.99 cm
Q11: Half of AC = 8 cm. h = √(17² − 8²) = √(289 − 64) = √225 = 15 cm
Q12: d = √(12² + 9²) = √(144 + 81) = √225 = 15 cm
Q13: h = √(10² − 6²) = √(100 − 36) = √64 = 8 m
Q14: BC = √(3² + 10²) = √(9 + 100) = √109 = 10.44 cm
Q15: D = √(8² + 5² + 3²) = √(64 + 25 + 9) = √98 = 9.90 cm
Q16: Half-diagonals: 6 cm and 4 cm. Side = √(6² + 4²) = √(36 + 16) = √52 = 7.21 cm
Q17: Horizontal offset = (20−12)/2 = 4 cm. h = √(10² − 4²) = √(100 − 16) = √84 = 9.17 cm
Q18: Other side = √(25.6² − 19.3²) = √(655.36 − 372.49) = √282.87 = 16.82 cm. Area = ½ × 19.3 × 16.82 = 162.50 cm²
Q19: 9² + 12² = 81 + 144 = 225. 15² = 225. Since 9² + 12² = 15², YES, right-angled at C.
Q20: (a) 7.2² + 9.6² = 51.84 + 92.16 = 144 = 12² → YES. (b) 14² + 48² = 196 + 2304 = 2500 = 50² → YES. (c) 5.5² + 7.3² = 30.25 + 53.29 = 83.54 ≠ 82.81 → NO.
Q21(a): Pos 1: √(13²−5²) = √144 = 12 m. Pos 2: √(13²−8²) = √105 = 10.25 m
Q21(b): 12 − 10.25 = 1.75 m
Q22(a): h = √(18² − 9²) = √(324 − 81) = √243 = 15.59 cm
Q22(b): Area = ½ × 18 × 15.59 = 140.30 cm²
Q23: x² + h² = 169, (20−x)² + h² = 225. Subtracting: (20−x)² − x² = 56 → 400−40x = 56 → x = 8.4. h = √(169 − 8.4²) = √97.24 = 9.86 cm
Q24(a): √(18² + 24²) = √(324 + 576) = √900 = 30 km
Q24(b): Ship 2: √(30² + 40²) = √(900 + 1600) = √2500 = 50 km. Ship 2 is further.
Q24(c): 30 km ÷ 12 km/h = 2.5 hours
Q25: Diagonal = √(24² + 18²) = √(576 + 324) = √900 = 30 cm ✓. Area = ½ × 24 × 18 = 216 cm²
Created by Miss Clarissa Ng · www.clartutors.com
Questions based on Cambridge IGCSE Mathematics (0580) syllabus · Source: Morrison & Hamshaw IGCSE Mathematics 2nd/3rd Edition