130. Pythagoras and Finding the Hypotenuse

Learning Intentions

• use Pythagoras’ theorem to find the hypotenuse of a right-angled triangle
• To understand what a surd is
• apply Pythagoras’ theorem to simple worded problems involving an unknown hypotenuse or diagonal

Pre-requisite Summary

• Know that a right-angled triangle has one angle of ${90}^{\circ }$
• Be able to identify the hypotenuse as the side opposite the right angle
• Recall that Pythagoras’ theorem is $a^2+b^2=c^2$ for a right-angled triangle
• Be able to square whole numbers and simple decimals
• Understand that the square root of a number is a value that, when multiplied by itself, gives the original number
• Know that not all square roots simplify to whole numbers
• Be able to interpret simple diagrams and worded geometry problems

Worked Examples

Worked Example 1

Find the hypotenuse of each right-angled triangle:
a) legs $3$ cm and $4$ cm
b) legs $5$ cm and $12$ cm

Worked Example 2

Find the hypotenuse of each right-angled triangle and give the exact answer first:
a) legs $6$ cm and $8$ cm
b) legs $7$ cm and $9$ cm

Worked Example 3

Write each answer as a surd, then as a decimal approximation where needed:
a) hypotenuse with legs $1$ cm and $1$ cm
b) hypotenuse with legs $2$ cm and $3$ cm

Worked Example 4

A rectangle has length $8$ cm and width $6$ cm. Find the length of its diagonal.

Worked Example 5

A ladder stands against a wall. The foot of the ladder is $9$ m from the wall and the wall height reached is $12$ m. Find the length of the ladder.

Worked Example 6

A television screen is rectangular with side lengths $24$ cm and $10$ cm. Find the screen diagonal. Give the exact answer and a decimal approximation.

Problems

Problem 1

Find the hypotenuse of each right-angled triangle:
a) legs $8$ cm and $15$ cm
b) legs $9$ cm and $12$ cm

Problem 2

Find the hypotenuse of each right-angled triangle and give the exact answer first:
a) legs $4$ cm and $7$ cm
b) legs $10$ cm and $11$ cm

Problem 3

Write each answer as a surd, then as a decimal approximation where needed:
a) hypotenuse with legs $1$ cm and $2$ cm
b) hypotenuse with legs $5$ cm and $6$ cm

Problem 4

A rectangle has length $12$ cm and width $5$ cm. Find the length of its diagonal.

Problem 5

A ladder stands against a wall. The foot of the ladder is $6$ m from the wall and the wall height reached is $8$ m. Find the length of the ladder.

Problem 6

A rectangular tablet has side lengths $18$ cm and $7$ cm. Find the screen diagonal. Give the exact answer and a decimal approximation.

Exercises

Understanding and Fluency

1. Find the hypotenuse of each right-angled triangle:
   a) legs $3$ cm and $4$ cm
   b) legs $8$ cm and $6$ cm
   c) legs $12$ cm and $5$ cm

2. Find the hypotenuse of each right-angled triangle:
   a) legs $7$ cm and $24$ cm
   b) legs $9$ cm and $40$ cm
   c) legs $11$ cm and $60$ cm

3. Find the hypotenuse and write the answer in simplest exact form:
   a) legs $1$ cm and $1$ cm
   b) legs $2$ cm and $5$ cm
   c) legs $4$ cm and $6$ cm

4. Find the hypotenuse and give a decimal approximation to $2$ decimal places:
   a) legs $5$ cm and $7$ cm
   b) legs $6$ cm and $11$ cm
   c) legs $8$ cm and $9$ cm

5. State whether each number is a surd or not a surd:
   a) $\sqrt{2}$
   b) $\sqrt{9}$
   c) $\sqrt{18}$
   d) $5$

6. Write each square root in simplest form if possible, and state whether it is a surd:
   a) $\sqrt{16}$
   b) $\sqrt{20}$
   c) $\sqrt{49}$
   d) $\sqrt{45}$

7. Find the diagonal of each rectangle:
   a) length $5$ cm, width $12$ cm
   b) length $9$ cm, width $9$ cm
   c) length $15$ cm, width $8$ cm

8. Solve each simple worded problem:
   a) A kite string forms a right-angled triangle with horizontal distance $10$ m and vertical height $24$ m. Find the string length.
   b) A ramp rises $1.2$ m over a horizontal distance of $1.6$ m. Find the ramp length.
   c) A door is $2$ m high and $0.9$ m wide. Find its diagonal.

Reasoning

9. Explain why the hypotenuse must be the side used as $c$ in $a^2+b^2=c^2$.

10. A student says that $\sqrt{25}$ is a surd. Explain the mistake.

11. Noah says that the hypotenuse of a right-angled triangle with sides $6$ cm and $8$ cm is ${10}^2=100$ cm because $6^2+8^2={10}^2$. Explain why this is incorrect.

12. Explain why a diagonal of a rectangle can be found using Pythagoras’ theorem.

13. A student writes the hypotenuse of a triangle with legs $2$ cm and $3$ cm as $\sqrt{5}$ cm. Describe the error.

Problem-solving

14. A rectangular garden is $7$ m long and $24$ m wide. Find the length of the diagonal path across the garden.

15. A ladder reaches a window $15$ m above the ground. If the ladder foot is $8$ m from the wall, how long is the ladder?

16. A square has side length $6$ cm. Find the diagonal of the square in exact form and as a decimal approximation.

17. A phone screen is $14$ cm long and $8$ cm wide. Find the length of the screen diagonal.

18. A sail forms a right-angled triangle with base $9$ m and height $12$ m. Find the length of the sloping side.

19. A rectangular painting is $45$ cm by $28$ cm. Find the diagonal length of the painting, giving the exact answer first.

Potential Misunderstandings

• Students may confuse the hypotenuse with one of the shorter sides
• Students may forget that the hypotenuse is opposite the right angle
• Students may substitute the side lengths into Pythagoras’ theorem incorrectly
• Students may add the side lengths instead of adding their squares
• Students may forget to take the square root after finding $a^2+b^2$
• Students may think any square root is a surd, even when it simplifies to a whole number
• Students may think a surd is just any decimal answer
• Students may give only a decimal approximation when an exact surd form is required
• Students may not recognise that diagonals of rectangles create right-angled triangles
• Students may ignore units when answering worded problems