GM Lesson 020: Pythagoras’ Theorem

Learning Intentions

• Understand Pythagoras’ theorem for right-angled triangles.
• Identify the hypotenuse and the two perpendicular sides.
• Use to represent the relationship between side lengths.

Prerequisites

• Recognise right angles in diagrams and practical contexts.
• Know that squaring a number means multiplying it by itself.
• Identify side lengths in a triangle.
• Substitute numbers into a formula.

Key Idea Summary

Pythagoras’ theorem applies only to right-angled triangles.

In a right-angled triangle:

• the hypotenuse is the side opposite the right angle;
• the hypotenuse is always the longest side;
• the other two sides are perpendicular sides.

The syllabus formula is:

where:

• is the length of the hypotenuse;
• and are the lengths of the two perpendicular sides.

This means:

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

Direct Instruction

Explain that Pythagoras’ theorem is a relationship between the three side lengths of a right-angled triangle.

Draw several right-angled triangles in different orientations. Emphasise that the hypotenuse is not always drawn horizontally or vertically. It is found by looking opposite the right angle.

For every right-angled triangle:

The letters and can be assigned to either perpendicular side, but must be the hypotenuse.

Worked Example 1: Identifying the Hypotenuse

A right-angled triangle has side lengths labelled , and . The right angle is between sides and .

Identify the hypotenuse.

Since the right angle is between and , the side opposite the right angle is .

Therefore, the hypotenuse is .

The Pythagoras relationship is:

Worked Example 2: Writing Pythagoras’ Theorem from a Diagram

A right-angled triangle has perpendicular sides cm and cm. The hypotenuse is cm.

Write the Pythagoras relationship.

The hypotenuse is , so .

The perpendicular sides are and .

Therefore:

So the square of the hypotenuse is .

Worked Example 3: Deciding Whether the Formula Has Been Written Correctly

A student writes the following for a right-angled triangle:

The triangle has side lengths , and , and the hypotenuse is .

Decide whether the equation is correct.

Since the hypotenuse is , the left side of the equation should be .

The correct equation is:

The student’s equation is incorrect because is not the hypotenuse.

Worked Example 4: Matching a Practical Situation to a Right-Angled Triangle

A ladder leans against a wall. The wall is vertical and the ground is horizontal. The ladder, wall and ground form a right-angled triangle.

The ladder is m long, the distance from the wall to the foot of the ladder is m, and the height reached on the wall is m.

Identify the hypotenuse and write the Pythagoras relationship.

The wall and ground meet at a right angle. The ladder is opposite the right angle, so the ladder is the hypotenuse.

Therefore, .

The perpendicular sides are and .

So:

Understanding Checks

Check 1

In a right-angled triangle, the right angle is between sides cm and cm. The third side is cm.

Which side is the hypotenuse?

Check 2

A right-angled triangle has hypotenuse and perpendicular sides and .

Write Pythagoras’ theorem for this triangle.

Check 3

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

Write the correct Pythagoras relationship.

Check 4

True or false:

In , the letter can represent any side of the triangle.

Explain your answer.

Exercises

Simple Familiar Exercises

Exercise 1

For each right-angled triangle, identify the hypotenuse.

a. The right angle is between sides cm and cm. The third side is cm.

b. The right angle is between sides and . The third side is .

c. The right angle is between sides m and m. The third side is m.

d. The right angle is between sides and . The third side is .

Exercise 2

For each triangle, write the Pythagoras relationship using .

a. Hypotenuse , perpendicular sides and .

b. Hypotenuse , perpendicular sides and .

c. Hypotenuse , perpendicular sides and .

d. Hypotenuse , perpendicular sides and .

Exercise 3

Decide whether each equation has the hypotenuse on the correct side. If not, rewrite it correctly.

a. A right-angled triangle has hypotenuse and perpendicular sides and .

b. A right-angled triangle has hypotenuse and perpendicular sides and .

c. A right-angled triangle has hypotenuse and perpendicular sides and .

d. A right-angled triangle has hypotenuse and perpendicular sides and .

Complex Familiar Exercises

Exercise 4

A rectangular park has length m and width m. A straight path is drawn from one corner of the park to the opposite corner.

a. Explain why this situation forms a right-angled triangle.

b. Identify the two perpendicular sides.

c. Identify the hypotenuse.

d. Write the Pythagoras relationship using for the diagonal path.

Exercise 5

A flagpole is supported by a cable. The cable runs from the top of the pole to a point on the ground. The pole is vertical and the ground is horizontal.

The pole has height m. The point on the ground is m from the base of the pole. The cable has length m.

a. Identify the right angle.

b. Identify the hypotenuse.

c. Write the Pythagoras relationship.

d. Explain why the cable must be longer than m.

Exercise 6

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

a. Identify the hypotenuse.

b. Write the Pythagoras relationship.

c. Substitute the side lengths into the relationship.

d. Check whether the relationship is true by evaluating both sides.

Homework Problems

Homework 1

For each right-angled triangle, identify the hypotenuse and write the Pythagoras relationship.

a. Perpendicular sides cm and cm; hypotenuse cm.

b. Perpendicular sides m and m; hypotenuse m.

c. Hypotenuse mm; perpendicular sides mm and mm.

d. Hypotenuse cm; perpendicular sides cm and cm.

Homework 2

A soccer field is rectangular. Its length is m and its width is m. A player runs diagonally from one corner to the opposite corner.

a. Draw a labelled right-angled triangle to represent the situation.

b. Identify the hypotenuse.

c. Write the Pythagoras relationship using for the diagonal distance.

Homework 3

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

a. Identify the hypotenuse.

b. Write the correct Pythagoras relationship.

c. Substitute the side lengths into the relationship.

d. Check whether the relationship is true.

Homework 4

A student writes:

for a right-angled triangle with side lengths , and .

a. Explain the mistake.

b. Write the correct Pythagoras relationship.

c. Explain how the hypotenuse can be identified from the side lengths.

Homework 5

A ramp, the ground and a vertical step form a right-angled triangle. The ramp is m long and the step is m high.

a. Identify the hypotenuse.

b. Identify the perpendicular sides.

c. Let represent the horizontal ground distance. Write the Pythagoras relationship.

d. Explain why the ramp length must be the longest side.

Next: GM Lesson 021 Finding Unknown Sides in Right Triangles