Year 9 Maths: Pythagoras’ theorem: find the missing leg length

What is Pythagoras’ Theorem?

Pythagoras’ theorem is a way to find the lengths of sides in a right-angled triangle. A right-angled triangle has one angle that is exactly 90 degrees.

The Formula

The formula is:

$$a^2 + b^2 = c^2$$

Where:

• ( a ) and ( b ) are the lengths of the two shorter sides (legs) of the triangle.
• ( c ) is the length of the longest side (hypotenuse), which is opposite the right angle.

Finding the Missing Leg Length

Sometimes, you might need to find one of the legs when you know the other leg and the hypotenuse. You can rearrange the formula to solve for ( a ) or ( b ):

1. To find ( a ): $$a^2 = c^2 – b^2$$
2. To find ( b ): $$b^2 = c^2 – a^2$$

Example 1

Question: Find the length of leg ( a ) if ( b = 3 ) cm and ( c = 5 ) cm.

Solution:

1. Use the formula for ( a ): $$a^2 = c^2 – b^2$$
2. Replace ( b ) and ( c ): $$a^2 = 5^2 – 3^2$$ $$a^2 = 25 – 9$$ $$a^2 = 16$$
3. Now, take the square root: $$a = 4 \text{ cm}$$

Tips and Tricks

• Draw a Diagram: Sketch the triangle and label the sides. This helps you visualise the problem.
• Check Your Work: After calculating, plug your answer back into the original equation to ensure it works.
• Units Matter: Always include units in your answers (cm, m, etc.).

Practice Questions

Easy Level (20 Questions)

1. Find ( a ) if ( b = 2 ) cm and ( c = 5 ) cm.
2. Find ( b ) if ( a = 1 ) cm and ( c = 2 ) cm.
3. Find ( a ) if ( b = 4 ) cm and ( c = 6 ) cm.
4. Find ( b ) if ( a = 3 ) cm and ( c = 5 ) cm.
5. Find ( a ) if ( b = 3 ) cm and ( c = 4 ) cm.
6. Find ( b ) if ( a = 5 ) cm and ( c = 13 ) cm.
7. Find ( a ) if ( b = 5 ) cm and ( c = 10 ) cm.
8. Find ( b ) if ( a = 6 ) cm and ( c = 10 ) cm.
9. Find ( a ) if ( b = 8 ) cm and ( c = 10 ) cm.
10. Find ( b ) if ( a = 0 ) cm and ( c = 5 ) cm.
11. Find ( a ) if ( b = 7 ) cm and ( c = 25 ) cm.
12. Find ( b ) if ( a = 12 ) cm and ( c = 13 ) cm.
13. Find ( a ) if ( b = 5 ) cm and ( c = 12 ) cm.
14. Find ( b ) if ( a = 8 ) cm and ( c = 10 ) cm.
15. Find ( a ) if ( b = 6 ) cm and ( c = 10 ) cm.
16. Find ( b ) if ( a = 9 ) cm and ( c = 15 ) cm.
17. Find ( a ) if ( b = 10 ) cm and ( c = 15 ) cm.
18. Find ( b ) if ( a = 7 ) cm and ( c = 25 ) cm.
19. Find ( a ) if ( b = 12 ) cm and ( c = 13 ) cm.
20. Find ( b ) if ( a = 5 ) cm and ( c = 7 ) cm.

Medium Level (20 Questions)

1. Find ( a ) if ( b = 6 ) cm and ( c = 10 ) cm.
2. Find ( b ) if ( a = 8 ) cm and ( c = 17 ) cm.
3. Find ( a ) if ( b = 12 ) cm and ( c = 20 ) cm.
4. Find ( b ) if ( a = 7 ) cm and ( c = 25 ) cm.
5. Find ( a ) if ( b = 15 ) cm and ( c = 17 ) cm.
6. Find ( b ) if ( a = 5 ) cm and ( c = 15 ) cm.
7. Find ( a ) if ( b = 9 ) cm and ( c = 11 ) cm.
8. Find ( b ) if ( a = 11 ) cm and ( c = 34 ) cm.
9. Find ( a ) if ( b = 13 ) cm and ( c = 15 ) cm.
10. Find ( b ) if ( a = 12 ) cm and ( c = 16 ) cm.
11. Find ( a ) if ( b = 14 ) cm and ( c = 18 ) cm.
12. Find ( b ) if ( a = 3 ) cm and ( c = 4 ) cm.
13. Find ( a ) if ( b = 5 ) cm and ( c = 7 ) cm.
14. Find ( b ) if ( a = 6 ) cm and ( c = 8 ) cm.
15. Find ( a ) if ( b = 10 ) cm and ( c = 12 ) cm.
16. Find ( b ) if ( a = 8 ) cm and ( c = 10 ) cm.
17. Find ( a ) if ( b = 15 ) cm and ( c = 19 ) cm.
18. Find ( b ) if ( a = 11 ) cm and ( c = 15 ) cm.
19. Find ( a ) if ( b = 8 ) cm and ( c = 15 ) cm.
20. Find ( b ) if ( a = 7 ) cm and ( c = 10 ) cm.

Hard Level (20 Questions)

1. Find ( a ) if ( b = 12 ) cm and ( c = 15 ) cm.
2. Find ( b ) if ( a = 15 ) cm and ( c = 20 ) cm.
3. Find ( a ) if ( b = 24 ) cm and ( c = 30 ) cm.
4. Find ( b ) if ( a = 9 ) cm and ( c = 12 ) cm.
5. Find ( a ) if ( b = 11 ) cm and ( c = 14 ) cm.
6. Find ( b ) if ( a = 8 ) cm and ( c = 12 ) cm.
7. Find ( a ) if ( b = 13 ) cm and ( c = 25 ) cm.
8. Find ( b ) if ( a = 7 ) cm and ( c = 10 ) cm.
9. Find ( a ) if ( b = 12 ) cm and ( c = 25 ) cm.
10. Find ( b ) if ( a = 15 ) cm and ( c = 17 ) cm.
11. Find ( a ) if ( b = 20 ) cm and ( c = 25 ) cm.
12. Find ( b ) if ( a = 35 ) cm and ( c = 37 ) cm.
13. Find ( a ) if ( b = 36 ) cm and ( c = 39 ) cm.
14. Find ( b ) if ( a = 40 ) cm and ( c = 42 ) cm.
15. Find ( a ) if ( b = 28 ) cm and ( c = 30 ) cm.
16. Find ( b ) if ( a = 33 ) cm and ( c = 35 ) cm.
17. Find ( a ) if ( b = 48 ) cm and ( c = 50 ) cm.
18. Find ( b ) if ( a = 5 ) cm and ( c = 7 ) cm.
19. Find ( a ) if ( b = 60 ) cm and ( c = 80 ) cm.
20. Find ( b ) if ( a = 45 ) cm and ( c = 60 ) cm.

Answers and Explanations

Easy Level Answers

1. 4 cm
2. 1.73 cm
3. 4 cm
4. 4 cm
5. 1 cm
6. 12 cm
7. 8.66 cm
8. 8 cm
9. 6 cm
10. 5 cm
11. 24 cm
12. 5 cm
13. 9.54 cm
14. 6 cm
15. 8 cm
16. 12 cm
17. 9 cm
18. 24.35 cm
19. 5 cm
20. 6.16 cm

Medium Level Answers

1. 8 cm
2. 15.13 cm
3. 16 cm
4. 24.22 cm
5. 8 cm
6. 14.87 cm
7. 6.32 cm
8. 32.57 cm
9. 6.4 cm
10. 12.65 cm
11. 10.77 cm
12. 3.94 cm
13. 5.83 cm
14. 7.21 cm
15. 10 cm
16. 8 cm
17. 9.22 cm
18. 12.41 cm
19. 10.44 cm
20. 8.06 cm

Hard Level Answers

1. 9 cm
2. 12.25 cm
3. 18 cm
4. 4.24 cm
5. 10.63 cm
6. 10.39 cm
7. 22.63 cm
8. 7.07 cm
9. 24.5 cm
10. 12.04 cm
11. 15 cm
12. 7.21 cm
13. 23.43 cm
14. 27.25 cm
15. 20.56 cm
16. 6.83 cm
17. 40 cm
18. 3.86 cm
19. 60 cm
20. 24.25 cm

Feel free to practice with these questions and remember to check your work! Happy learning!