GM Lesson 022 Practical Problems in Two Dimensions

Learning Intentions

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

• Use Pythagoras’ theorem to solve practical two-dimensional problems.
• Represent practical situations using right-angled triangle diagrams.
• Interpret calculated lengths using appropriate units.

Prerequisites

Students should already be able to:

• Identify the hypotenuse in a right-angled triangle.

• Use Pythagoras’ theorem:

• Calculate the hypotenuse of a right-angled triangle.

• Calculate a shorter side of a right-angled triangle.

• Round decimal answers appropriately.

• Attach correct units to length measurements.

Key Idea Summary

Pythagoras’ theorem can be used when a practical situation can be represented by a right-angled triangle.

If the unknown side is the hypotenuse, use:

or

If the unknown side is one of the shorter sides, rearrange:

or

The longest side must always be the hypotenuse. In practical problems, the answer should include suitable units such as metres, centimetres or kilometres.

Direct Instruction and Worked Examples

Teacher Notes for Timing

• Learning intentions and review: minutes
• Direct instruction and worked examples: minutes
• Understanding checks: minutes
• Exercises: minutes

Direct Instruction

To solve practical two-dimensional problems using Pythagoras’ theorem:

1. Read the situation carefully.
2. Draw a right-angled triangle diagram.
3. Label the known lengths.
4. Decide whether the unknown length is the hypotenuse or a shorter side.
5. Substitute into Pythagoras’ theorem.
6. Calculate the unknown length.
7. Interpret the answer with appropriate units.

Worked Example 1: Finding a Diagonal Distance

A rectangular sports field is m long and m wide. Find the diagonal distance from one corner to the opposite corner.

The situation forms a right-angled triangle where the two shorter sides are m and m.

Let the diagonal be .

Therefore, the diagonal distance is m.

Worked Example 2: A Ladder Against a Wall

A ladder is m long and rests against a vertical wall. The base of the ladder is m from the wall. How high up the wall does the ladder reach?

The ladder is the hypotenuse because it is opposite the right angle and is the longest side.

Let the height up the wall be .

Therefore, the ladder reaches m up the wall.

Worked Example 3: Shortest Walking Distance

A park has two straight paths that meet at right angles. One path is m long and the other is m long. A new straight path is built directly between the two far ends. Find the length of the new path.

The new path is the hypotenuse.

Let the new path have length .

Therefore, the new straight path is m long.

Understanding Checks

Check 1

A rectangle has length cm and width cm.

1. Which length would be the hypotenuse if the diagonal is drawn?
2. Write the Pythagoras equation needed to find the diagonal.
3. Calculate the diagonal length.

Check 2

A m ladder reaches m up a wall.

1. Is the ladder a shorter side or the hypotenuse?
2. Write the equation needed to find the distance from the wall to the base of the ladder.
3. Calculate the distance from the wall to the base of the ladder.

Check 3

A student writes:

1. What practical situation could this equation represent?
2. Is the hypotenuse or a shorter side?
3. Calculate .

Exercises

Simple Familiar Exercises

Exercise 1

A rectangle is cm long and cm wide. Find the length of its diagonal.

Exercise 2

A right-angled garden bed has perpendicular sides of m and m. Find the length of the diagonal edge.

Exercise 3

A ramp rises m over a horizontal distance of m. Find the length of the ramp, correct to decimal place.

Exercise 4

A television screen is cm wide and cm high. Find the diagonal screen size.

Complex Familiar Exercises

Exercise 5

A ladder is m long. Its base is placed m from a wall. How high up the wall does the ladder reach?

Exercise 6

A rectangular room is m long and m wide. Find the distance from one corner of the room to the opposite corner, correct to decimal places.

Exercise 7

A diagonal support beam is used across a rectangular frame. The frame is m high and m wide. Find the length of the support beam.

Exercise 8

A cyclist travels km east and then km north. Find the cyclist’s straight-line distance from the starting point.

Homework Problems

Problem 1

A rectangle has length cm and width cm. Find the diagonal length.

Problem 2

A ladder is m long and reaches m up a wall. Find the distance between the base of the ladder and the wall.

Problem 3

A soccer field is m long and m wide. Find the diagonal distance from one corner to the opposite corner, correct to the nearest metre.

Problem 4

A hiker walks km south and then km west. Find the hiker’s straight-line distance from the starting point.

Problem 5

A rectangular picture frame is cm wide and cm high. A diagonal brace is added from one corner to the opposite corner. Find the length of the brace.

Problem 6

A ramp is built to rise m over a horizontal distance of m. Find the length of the ramp, correct to decimal places.

Problem 7

A park is shaped like a rectangle with dimensions m by m. A direct walking track is built diagonally across the park. Find the track length, correct to the nearest metre.

Problem 8

A flagpole is supported by a wire from the top of the pole to a point on the ground m from the base. If the pole is m tall, find the length of the wire.

Next: GM Lesson 023 Simple Applications in Three Dimensions