131. Pythagoras and Finding a Shorter Side

Learning Intentions

• use Pythagoras’ theorem to find the length of a shorter side in a right-angled triangle
• apply Pythagoras’ theorem to simple worded problems involving an unknown shorter side

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 $a^2+b^2=c^2$
• Be able to square whole numbers and simple decimals
• Be able to find square roots using a calculator where needed
• Understand that to find a shorter side, one squared side is found by subtracting from the square of the hypotenuse
• Be able to interpret simple diagrams and worded geometry problems

Worked Examples

Worked Example 1

Find the shorter side of each right-angled triangle:
a) hypotenuse $=13$ cm, other shorter side $=5$ cm
b) hypotenuse $=10$ cm, other shorter side $=6$ cm

Worked Example 2

Find the shorter side of each right-angled triangle:
a) hypotenuse $=17$ cm, other shorter side $=8$ cm
b) hypotenuse $=25$ cm, other shorter side $=7$ cm

Worked Example 3

Find the shorter side and give the exact answer first where needed:
a) hypotenuse $=5$ cm, other shorter side $=3$ cm
b) hypotenuse $=10$ cm, other shorter side $=7$ cm

Worked Example 4

A ladder of length $13$ m reaches a wall. The foot of the ladder is $5$ m from the wall. Find the height reached on the wall.

Worked Example 5

A rectangle has diagonal $15$ cm and length $12$ cm. Find the width.

Worked Example 6

A ramp has length $7.5$ m and rises $4.5$ m vertically. Find the horizontal distance covered by the ramp.

Problems

Problem 1

Find the shorter side of each right-angled triangle:
a) hypotenuse $=15$ cm, other shorter side $=9$ cm
b) hypotenuse $=26$ cm, other shorter side $=10$ cm

Problem 2

Find the shorter side of each right-angled triangle:
a) hypotenuse $=20$ cm, other shorter side $=12$ cm
b) hypotenuse $=29$ cm, other shorter side $=21$ cm

Problem 3

Find the shorter side and give the exact answer first where needed:
a) hypotenuse $=8$ cm, other shorter side $=4$ cm
b) hypotenuse $=13$ cm, other shorter side $=9$ cm

Problem 4

A ladder of length $10$ m reaches a wall. The foot of the ladder is $6$ m from the wall. Find the height reached on the wall.

Problem 5

A rectangle has diagonal $17$ cm and length $15$ cm. Find the width.

Problem 6

A ramp has length $6.5$ m and rises $2.5$ m vertically. Find the horizontal distance covered by the ramp.

Exercises

Understanding and Fluency

1. Complete each statement:
   a) To find a shorter side in a right-angled triangle, subtract the square of the known shorter side from the square of the ______
   b) In $a^2+b^2=c^2$, the letter $c$ represents the ______
   c) After subtracting the squares, take the ______ root to find the missing side

2. Find the missing shorter side:
   a) hypotenuse $=5$ cm, other shorter side $=4$ cm
   b) hypotenuse $=13$ cm, other shorter side $=12$ cm
   c) hypotenuse $=17$ cm, other shorter side $=15$ cm

3. Find the missing shorter side:
   a) hypotenuse $=25$ cm, other shorter side $=24$ cm
   b) hypotenuse $=10$ cm, other shorter side $=8$ cm
   c) hypotenuse $=41$ cm, other shorter side $=9$ cm

4. Find the missing shorter side and give the exact answer first where needed:
   a) hypotenuse $=6$ cm, other shorter side $=2$ cm
   b) hypotenuse $=10$ cm, other shorter side $=6$ cm
   c) hypotenuse $=15$ cm, other shorter side $=11$ cm

5. Find the missing shorter side and give a decimal approximation to $2$ decimal places:
   a) hypotenuse $=9$ cm, other shorter side $=4$ cm
   b) hypotenuse $=12$ cm, other shorter side $=5$ cm
   c) hypotenuse $=20$ cm, other shorter side $=13$ cm

6. Find the width of each rectangle:
   a) diagonal $=13$ cm, length $=5$ cm
   b) diagonal $=25$ cm, length $=24$ cm
   c) diagonal $=10$ cm, length $=6$ cm

7. Solve each simple worded problem:
   a) A ladder is $15$ m long and stands $9$ m from a wall. How high up the wall does it reach?
   b) A tent support is $13$ m long and reaches a point $5$ m above the ground. How far is its base from that point directly below?
   c) A television screen has diagonal $20$ cm and width $16$ cm. Find its height.

8. Solve each simple worded problem:
   a) A ramp is $8$ m long and rises $3$ m. Find the horizontal distance.
   b) A rectangular garden has diagonal $25$ m and one side $7$ m. Find the other side.
   c) A kite string is $10$ m long and the kite is $8$ m above the ground. Find the horizontal distance from the person to the point directly below the kite.

Reasoning

9. Explain why the hypotenuse cannot be the side you are solving for when using subtraction to find a shorter side.

10. A student says that if the hypotenuse is $13$ cm and one shorter side is $5$ cm, then the other shorter side is $\sqrt{{13}^2+5^2}$. Explain the mistake.

11. Noah says that the missing shorter side in a triangle with hypotenuse $10$ cm and one side $6$ cm is $10-6=4$ cm. Is he correct? Explain.

12. Explain why you must square the side lengths before subtracting.

13. A student writes that for a triangle with hypotenuse $17$ cm and one side $8$ cm, the missing side is $\sqrt{17-8}$. Describe the error.

Problem-solving

14. A ladder is $17$ m long and its foot is $8$ m from a wall. Find the height reached on the wall.

15. A rectangular poster has diagonal $15$ cm and one side $9$ cm. Find the other side.

16. A ramp is $10$ m long and rises $6$ m vertically. Find the horizontal distance.

17. A wire runs from the top of a pole to the ground. The wire is $13$ m long and the base of the pole is $5$ m from the anchor point. Find the height of the pole.

18. A rectangular floor tile has diagonal $26$ cm and one side $10$ cm. Find the other side.

19. A kite string is $25$ m long and the kite is $24$ m above the ground. Find the horizontal distance from the flyer to the point directly below the kite.

Potential Misunderstandings

• Students may subtract the side lengths instead of subtracting their squares
• Students may forget that the hypotenuse is the longest side
• Students may use the wrong side as the hypotenuse
• Students may add the squares instead of subtracting when finding a shorter side
• Students may forget to take the square root after subtracting
• Students may try to subtract before squaring
• Students may confuse finding a shorter side with finding the hypotenuse
• Students may ignore units in worded problems
• Students may not recognise right-angled triangles hidden inside rectangles or practical situations