GM Lesson 021 Finding Unknown Sides in Right Triangles

Learning Intentions

By the end of this lesson, students will be able to:

• Calculate the hypotenuse of a right-angled triangle.
• Calculate a shorter side of a right-angled triangle.
• Check whether an answer is reasonable using the relative side lengths.

Prerequisites

Students should already be able to:

• Identify the hypotenuse as the side opposite the right angle.
• Identify the two perpendicular sides of a right-angled triangle.
• Substitute values into the formula .
• Find square roots using a calculator.
• Round decimal answers appropriately.

Key Idea Summary

Pythagoras’ theorem applies only to right-angled triangles.

If is the hypotenuse and and are the two shorter perpendicular sides, then:

To find the hypotenuse:

To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse:

or

The hypotenuse must always be the longest side. Therefore, if a calculated shorter side is longer than the hypotenuse, the answer is unreasonable.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Step 1: Identify the Hypotenuse

In a right-angled triangle:

• the hypotenuse is opposite the right angle
• the hypotenuse is the longest side
• the shorter sides meet at the right angle

Before using Pythagoras’ theorem, label the hypotenuse as .

Step 2: Decide What Is Unknown

There are two main cases.

If the hypotenuse is unknown, use:

If a shorter side is unknown, use:

Worked Example 1: Finding the Hypotenuse

A right-angled triangle has shorter sides of length cm and cm. Find the length of the hypotenuse.

Since the hypotenuse is unknown, use:

Substitute and :

The hypotenuse is cm.

Reasonableness check:

Since the hypotenuse is longer than both shorter sides, the answer is reasonable.

Worked Example 2: Finding the Hypotenuse with a Decimal Answer

A right-angled triangle has shorter sides of length m and m. Find the hypotenuse correct to decimal place.

Correct to decimal place:

The hypotenuse is approximately m.

Reasonableness check:

The hypotenuse is longer than both shorter sides, so the answer is reasonable.

Worked Example 3: Finding a Shorter Side

A right-angled triangle has hypotenuse cm and one shorter side cm. Find the other shorter side.

Since a shorter side is unknown, subtract the square of the known shorter side from the square of the hypotenuse.

Let the unknown shorter side be .

Substitute and :

The unknown shorter side is cm.

Reasonableness check:

The shorter side is less than the hypotenuse, so the answer is reasonable.

Worked Example 4: Finding a Shorter Side in Context

A ladder is m long and reaches a point m up a wall. The ground, wall and ladder form a right-angled triangle. How far is the foot of the ladder from the wall?

The ladder is the hypotenuse because it is opposite the right angle.

Let be the distance from the wall.

The foot of the ladder is m from the wall.

Reasonableness check:

The distance from the wall is shorter than the ladder, so the answer is reasonable.

Understanding Checks

Check 1

In a right-angled triangle, the shorter sides are cm and cm.

1. Which side is unknown?
2. Should you add or subtract the squares?
3. Write the correct substitution into .

Check 2

A right-angled triangle has hypotenuse m and one shorter side m.

1. Which side must be labelled ?
2. Should the calculation involve or ?
3. Explain why.

Check 3

A student calculates that a right-angled triangle has hypotenuse cm and a shorter side cm.

Explain why this answer cannot be correct.

Check 4

A right-angled triangle has side lengths cm, cm and cm.

Use the relationship between side lengths to check whether this could be a right-angled triangle.

Exercises

Simple Familiar Exercises

Exercise 1

Find the hypotenuse of a right-angled triangle with shorter sides cm and cm.

Exercise 2

Find the hypotenuse of a right-angled triangle with shorter sides m and m.

Exercise 3

Find the hypotenuse of a right-angled triangle with shorter sides cm and cm.

Exercise 4

Find the unknown shorter side of a right-angled triangle with hypotenuse cm and one shorter side cm.

Exercise 5

Find the unknown shorter side of a right-angled triangle with hypotenuse m and one shorter side m.

Exercise 6

Find the unknown shorter side of a right-angled triangle with hypotenuse cm and one shorter side cm.

Complex Familiar Exercises

Exercise 7

A rectangular park is m long and m wide. A path runs diagonally across the park from one corner to the opposite corner.

Find the length of the diagonal path.

Exercise 8

A television screen is cm wide and cm high.

Find the diagonal length of the screen correct to decimal place.

Exercise 9

A ladder is m long and reaches m up a wall.

Find how far the base of the ladder is from the wall.

Exercise 10

A right-angled triangle has hypotenuse cm and one shorter side cm.

Find the other shorter side correct to decimal place.

Exercise 11

A soccer field is m long and m wide.

Find the distance from one corner of the field to the opposite corner correct to the nearest metre.

Exercise 12

A ramp is m long and rises m vertically.

Find the horizontal distance covered by the ramp correct to decimal places.

Homework Problems

Problem 1

Find the hypotenuse of a right-angled triangle with shorter sides cm and cm.

Problem 2

Find the hypotenuse of a right-angled triangle with shorter sides m and m. Round to decimal place.

Problem 3

Find the unknown shorter side of a right-angled triangle with hypotenuse cm and one shorter side cm.

Problem 4

Find the unknown shorter side of a right-angled triangle with hypotenuse m and one shorter side m. Round to decimal place.

Problem 5

A ladder is m long and reaches m up a wall.

Find how far the base of the ladder is from the wall. Round to decimal places.

Problem 6

A rectangular block has a front face measuring cm by cm.

Find the diagonal length across the front face.

Problem 7

A boat travels km north and then km east.

Find the direct distance from its starting point.

Problem 8

A right-angled triangle has side lengths cm, cm and cm.

Check whether these measurements form a right-angled triangle.

Problem 9

A rectangular phone screen has diagonal length cm and height cm.

Find the width of the screen correct to decimal place.

Problem 10

A student finds the missing shorter side of a right-angled triangle with hypotenuse cm and one shorter side cm. Their answer is cm.

Explain why the answer is unreasonable, then describe the correct method.

Next: GM Lesson 022 Practical Problems in Two Dimensions