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E2.4 · Model and apply the Pythagorean relationship in right triangles

Learn to model and apply the pythagorean relationship in right triangles through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Spatial Sense

Model and apply the relationship in right triangles

A ladder leaning against a wall makes a triangle with the wall and the ground. The wall and ground meet at a right angle. The ladder is across from that angle. This kind of triangle lets us use the Pythagorean relationship to find a missing length. A right angle measures 90°. A right triangle is a triangle with one right angle. In this lesson, we will use drawings, square areas, and calculations to find missing side lengths.

What you will learn

  • Identify the legs and hypotenuse in a right triangle.
  • Explain the relationship using the areas of squares.
  • Find a missing side length when the other two side lengths are known.
  • Check whether a calculated side length is reasonable.

Build the relationship from squares

The two sides that meet to make the right angle are called the legs. The side across from the right angle is called the hypotenuse. The hypotenuse is the longest side of a right triangle.
Imagine drawing a square along each side of a right triangle. A square’s area is found by multiplying its side length by itself. For example, a square with side length 4 units has area 16 square units, because 4×4=164\times4=16.
For a right triangle with legs of 3 units and 4 units, the squares on the legs have areas of 9 and 16 square units. Their total is 25 square units. A square with area 25 square units has side length 5 units. The square on the hypotenuse has that same area.
This area pattern is the Pythagorean relationship: the areas of the two squares on the legs add to the area of the square on the hypotenuse. It applies to right triangles. First find the right angle, then identify the hypotenuse across from it.
a2+b2=c2a^2+b^2=c^2
  • The legs meet at the right angle.
  • The hypotenuse is across from the right angle.
  • The two leg-square areas add to the hypotenuse-square area.

Choose the calculation that fits

The raised 2 in a number such as 626^2 means “squared.” Multiply the number by itself: 62=6×6=366^2=6\times6=36. The result is a square value, not a side length.
When the two legs are known, square each leg length and add the square values. The total is the hypotenuse’s square. Find the positive side length whose square equals that total.
When the hypotenuse and one leg are known, subtract the known leg’s square from the hypotenuse’s square. The result is the missing leg’s square. Then identify the positive length whose square equals that result. For example, if the missing leg’s square is 144, the length is 12 because 122=14412^2=144.
Use the same unit for all side lengths. A side length describes a distance, so it is positive. The hypotenuse must be longer than either leg. These checks help you notice a mix-up or calculation error.
c2=a2+b2c^2=a^2+b^2
  • For two known legs, add their squared lengths.
  • For a known hypotenuse and leg, subtract the leg’s square from the hypotenuse’s square.
  • A square value is not the same as a side length.

Read a diagram carefully

A right-triangle diagram shows how the sides meet. The small square at a corner marks the right angle. The sides that meet there are the legs. The side opposite that corner is the hypotenuse.
A drawing may not be made to scale. Do not choose a side because it looks longest or shortest. Use the right-angle mark and the measurements stated in the problem. A labelled diagram can help organize the information, but the relationship—not the apparent size in the picture—determines the answer.
  • Find the right-angle mark before naming the sides.
  • The hypotenuse is opposite the right angle.
  • Use the given measurements rather than estimating from the drawing.

Check the result

After finding a missing length, put the three side lengths into the relationship. The two leg-square values should add to the hypotenuse-square value. If they do not, revisit the calculation.
Check the side lengths too. The hypotenuse must be longer than each leg. If a calculated leg is longer than the hypotenuse, check that you identified the right angle and used subtraction for a missing leg.
  • Check the square values in the relationship.
  • Check that the hypotenuse is the longest side.

Worked example

Find the hypotenuse

A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
Right triangle with legs 6 cm and 8 cm
Right triangle with legs 6 cm and 8 cmThe two perpendicular legs are 6 cm and 8 cm. The hypotenuse is calculated from their squares.6 cm8 cm10 cm6² + 8² = 100

The two perpendicular legs are 6 cm and 8 cm. The hypotenuse is calculated from their squares.

  1. Identify the sides
    The two given sides meet at the right angle, so they are the legs. The missing side is across from that angle, so it is the hypotenuse.
  2. Square and add
    Square each leg length and add the results. The total is the hypotenuse’s square.
    62+82=36+64=1006^2+8^2=36+64=100
  3. Find the length
    The positive length whose square is 100 is 10. The hypotenuse is 10 cm, which is longer than either leg.
    102=10010^2=100
Answer: The hypotenuse is 10 cm.
Check: The relationship is satisfied: 62+82=1026^2+8^2=10^2. The hypotenuse is longer than both legs.

Worked example

Find a missing leg

A right triangle has a hypotenuse of 13 m and one leg of 5 m. Find the other leg.
Right triangle with hypotenuse 13 m
Right triangle with hypotenuse 13 mOne perpendicular leg is 5 m. The other leg is calculated from the hypotenuse and known leg.5 m12 m13 m5² + 12² = 169

One perpendicular leg is 5 m. The other leg is calculated from the hypotenuse and known leg.

  1. Identify the side roles
    The 13 m side is the hypotenuse. The 5 m side is one leg. Subtract the known leg’s square from the hypotenuse’s square to get the missing leg’s square.
  2. Find the missing leg’s square
    Square the two known lengths, then subtract. This gives 144 for the missing leg’s square.
    132−52=169−25=14413^2-5^2=169-25=144
  3. Find the side length
    The positive length whose square is 144 is 12, because 12 multiplied by 12 is 144. The missing leg is 12 m, shorter than the 13 m hypotenuse.
    122=14412^2=144
Answer: The other leg is 12 m.
Check: The two leg squares add to the hypotenuse square: 52+122=1325^2+12^2=13^2.

Worked example

Find a diagonal path

A rectangular garden is 9 m long and 12 m wide. A straight path runs from one corner to the opposite corner. How long is the path?
Garden sides and diagonal path
Garden sides and diagonal pathThe garden sides of 9 m and 12 m meet at a right angle. The diagonal path is the hypotenuse.9 m12 m15 m9² + 12² = 225

The garden sides of 9 m and 12 m meet at a right angle. The diagonal path is the hypotenuse.

  1. Make a triangle model
    The garden’s length and width meet at a right angle. They are the legs of a right triangle. The diagonal path is across from the right angle, so it is the hypotenuse.
  2. Square and add
    Use the two garden sides as the legs. Their squared lengths add to the path’s squared length.
    92+122=81+144=2259^2+12^2=81+144=225
  3. Find and check the path length
    The positive length whose square is 225 is 15. The path is 15 m long. It is longer than either garden side, as a hypotenuse should be.
    152=22515^2=225
Answer: The diagonal path is 15 m long.
Check: The relationship is satisfied: 92+122=1529^2+12^2=15^2.

Common mistakes and how to avoid them

Using the relationship without checking for a right angle.
Correction: Use the Pythagorean relationship only for a right triangle.
Calling a side the hypotenuse because it looks longest in the drawing.
Correction: The hypotenuse is across from the right angle. The drawing may not be to scale.
Adding side lengths instead of their squares.
Correction: Square each leg length first, then add the square values.
Reporting a square value as a side length.
Correction: If the missing side’s square is 144, the side length is 12, not 144.

Lesson summary

  • The legs of a right triangle meet at the right angle.
  • The hypotenuse is opposite the right angle and is the longest side.
  • The two leg-square values add to the hypotenuse-square value.
  • For a missing leg, subtract the known leg’s square from the hypotenuse’s square, then find the positive length with that square value.
  • Check the relationship and make sure the hypotenuse is longest.

Check your understanding

Question 1

A right triangle has legs of 7 cm and 24 cm. What is its hypotenuse?
  1. 25 cm
  2. 31 cm
  3. 17 cm
  4. 576 cm
Show answer and explanation
25 cm
The leg squares total 72+242=49+576=6257^2+24^2=49+576=625. Since 252=62525^2=625, the hypotenuse is 25 cm.

Question 2

A right triangle has a hypotenuse of 10 m and one leg of 6 m. What is the other leg?
  1. 4 m
  2. 8 m
  3. 16 m
  4. 64 m
Show answer and explanation
8 m
The missing leg’s square is 102−62=100−36=6410^2-6^2=100-36=64. The positive length whose square is 64 is 8 m.

Key terms

Right angle
An angle that measures 90°.
Right triangle
A triangle with one right angle.
Leg
Either of the two sides that meet at the right angle.
Hypotenuse
The side across from the right angle; it is the longest side of a right triangle.
Square a number
Multiply the number by itself.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation E2.4. It is a study resource, not an official curriculum publication.

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