Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

The Pythagorean Relationship

E2.4 A1About 50 minutes
Learning goals
  • I can name the legs and the hypotenuse of a right triangle.
  • I can describe the Pythagorean relationship using squares drawn on the sides of a right triangle.
  • I can write the relationship as a² + b² = c².
  • I can use the relationship to test whether a triangle has a right angle.

Words to know

right triangle
A triangle with one 90° angle.
legs
The two sides of a right triangle that form the right angle. Often called a and b.
hypotenuse
The side across from the right angle. It is always the longest side. Often called c.
Pythagorean relationship
In every right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs: a² + b² = c². Also called the Pythagorean theorem.
square root
The number that multiplies by itself to make a given number. √25 = 5 because 5 × 5 = 25.
Pythagorean triple
Three whole numbers that fit a² + b² = c², like 3, 4, 5.

Remember from Grade 7

In Grade 7 you learned that a square number is a number times itself (5² = 25), and that the square root undoes squaring: √25 = 5. You also know that the area of a square with side s is s². Today these ideas come together in one of the most famous relationships in mathematics.

Let's start with a carpenter's trick

Builders need the corners of decks, sheds and rooms to be perfectly square (90°). A common trick: measure 3 units along one edge and 4 units along the other, then check that the diagonal between those two marks is exactly 5 units. If it is, the corner is square. Why does 3, 4, 5 work? The answer is the Pythagorean relationship, named after the ancient Greek mathematician Pythagoras. Other cultures, including mathematicians in ancient Babylon, India and China, knew about this relationship too.

91625legs: 3 and 4hypotenuse: 53² + 4² = 5²9 + 16 = 25
A right triangle with legs 3 and 4 units and hypotenuse 5 units, drawn to scale, with a square on each side. The squares on the legs hold 9 and 16 unit squares. Together that is 25, exactly the area of the square on the hypotenuse.

The relationship

In a right triangle with legs a and b and hypotenuse c:

a² + b² = c²

In words: the area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs.

Important things to notice:

  • It works for every right triangle, and only for right triangles.
  • c must be the hypotenuse: the longest side, across from the right angle.
  • It is about squares of the sides, not the sides themselves: 3 + 4 is not 5, but 3² + 4² = 5².

Why it works: a rearranging model

Draw a big square with sides of length a + b. Inside it, place four copies of the right triangle in two different ways:

  1. Put a triangle in each corner so their hypotenuses make a tilted square in the middle. The empty space is c².
  2. Pair the triangles into two rectangles, a by b, in opposite corners. The empty space is two squares: a² and b².

The big square and the four triangles are the same both times, so the empty spaces must be equal: a² + b² = c².

c²abbaa²b²abableftover: c²leftover: a² + b²
The rearranging model with a = 3 and b = 4. Both big squares have sides a + b and hold the same four yellow triangles. So the blue leftover areas must be equal: c² = a² + b².

Testing for a right angle

The relationship also works backwards. For any triangle, call the longest side c and compare:

  • a² + b² = c²: the triangle is a right triangle.
  • a² + b² > c²: the largest angle is less than 90° (acute).
  • a² + b² < c²: the largest angle is more than 90° (obtuse).
Worked example

Example 1: Check a right triangle

A triangle has sides of 5 cm, 12 cm and 13 cm. Is it a right triangle?

  1. The longest side is 13 cm, so c = 13.
  2. a² + b² = 5² + 12² = 25 + 144 = 169.
  3. c² = 13² = 169.
  4. They are equal.

Yes. 5² + 12² = 13², so the angle across from the 13 cm side is 90°.

Worked example

Example 2: Not a right triangle

A triangle has sides of 6, 7 and 9. Is it a right triangle? If not, is its largest angle more or less than 90°?

  1. c = 9 (the longest side).
  2. 6² + 7² = 36 + 49 = 85. 9² = 81.
  3. 85 is not equal to 81, so it is not a right triangle.
  4. 85 > 81: the two shorter sides are "too long" for a right angle, so the angle between them is less than 90°.

No. Its largest angle is less than 90° (acute).

Worked example

Example 3: The carpenter's check

A deck frame has two sides meeting at a corner. A builder marks 60 cm along one side and 80 cm along the other. What should the diagonal between the marks measure if the corner is square?

  1. 60² + 80² = 3 600 + 6 400 = 10 000.
  2. The diagonal squared must be 10 000. √10 000 = 100, because 100 × 100 = 10 000.

100 cm. (It's the 3-4-5 triangle with every side multiplied by 20.)

Think like a mathematician

  • Always find the hypotenuse first: it's across from the right angle and it's the longest side.
  • Write the relationship, then substitute: a² + b² = c².
  • Learn a few Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples work too: 6-8-10, 9-12-15, 30-40-50.
Hands-on activity

Squares on the sides

You need: graph paper (or paper you rule into 1 cm squares), scissors (a grown-up can help), a ruler, tape. Optional: a long piece of string and painter's tape for the floor.

  1. Cut out three squares of graph paper: 3 by 3, 4 by 4 and 5 by 5.
  2. Arrange them so their corners touch and their edges make a triangle in the middle. You've made a right triangle! Check the corner with the corner of a sheet of paper.
  3. Cut the 3 by 3 and 4 by 4 squares into single unit squares. Can you fit all 25 of them exactly onto the 5 by 5 square?
  4. Now try squares of 4, 5 and 6. Arrange them into a triangle. Is the angle between the two smaller squares a right angle? Compare 4² + 5² with 6².
  5. Floor version: with painter's tape, mark a corner of a room or a rug. Measure 30 cm along one edge and 40 cm along the other. Is the diagonal 50 cm? If so, the corner is square.

What to look for: 9 + 16 fills 25 exactly. For 4, 5, 6: 16 + 25 = 41, which is more than 36, so the angle is less than 90°.

Math talk: "Why do we compare areas of squares, not just the side lengths?"

Movie math mistake

In the 1939 movie The Wizard of Oz, the Scarecrow gets a diploma and proudly recites a version of the Pythagorean theorem. He gets it wrong! He talks about the square roots of the sides of an isosceles triangle. Can you spot what he should have said?

Your turn

1

A right triangle has sides of 8, 15 and 17. Which side is the hypotenuse?

2

Check that 8, 15, 17 is a right triangle.

3

Is a triangle with sides 4, 5, 6 a right triangle?

4

The squares on the legs of a right triangle have areas of 36 cm² and 64 cm². What is the area of the square on the hypotenuse? How long is the hypotenuse?

5

The square on the hypotenuse has an area of 169 cm², and the square on one leg has an area of 25 cm². Find the area of the square on the other leg, and that leg's length.

6

A right triangle has legs p and q and hypotenuse r. Write the Pythagorean relationship.

7

Is 9, 12, 15 a right triangle?

More practice: warm-up

1

What is 7²?

2

What is √81?

3

The hypotenuse is across from which angle?

4

What is 3² + 4²?

5

Which side of a right triangle is always the longest?

More practice: on your own

1

Is 7, 24, 25 a right triangle?

2

A triangle has sides of 10, 10 and 14. Is it a right triangle? If not, is its largest angle more or less than 90°?

3

A builder measures 1.2 m and 1.6 m along the two sides of a corner, and the diagonal between the marks is 2 m. Is the corner square?

4

The squares on the legs have areas of 20 cm² and 29 cm². How long is the hypotenuse?

5

Is 1.5, 2, 2.5 a right triangle?

Challenge

1

Explain why 6, 8, 10 is a right triangle if you already know 3, 4, 5 is, using the areas of the squares.

2

Classify the triangle with sides 5, 6, 8 as right, acute or obtuse.

3

Describe how you could use four copies of a right triangle inside a square with sides a + b to show that a² + b² = c².

Check your understanding

Quiz

1.

The hypotenuse of a right triangle is:

2.

Which set of side lengths makes a right triangle?

3.

The squares on the legs of a right triangle have areas of 9 and 16. What is the area of the square on the hypotenuse?

4.

The relationship a² + b² = c² is true for:

5.

A right triangle has legs of 6 and 8. What is a² + b²?

6.

A builder measures 3 m and 4 m along two sides of a corner, and the diagonal is 5.2 m. What does that tell her?

For parents and teachers

Big idea: The Ontario curriculum (E2.4) asks Grade 8 students to describe the Pythagorean relationship using various geometric models, and apply it to find an unknown side (the next lesson). This lesson focuses on the models: squares on the sides, and the rearranging argument.

Common difficulties:

  • Adding the sides instead of the squares (thinking 3 + 4 = 5).
  • Using the wrong side as c. The hypotenuse is always across from the right angle and is the longest side.
  • Applying the relationship to triangles that aren't right triangles.

How to help: The 3-4-5 check is used by real builders and gardeners. Try it together on a rug, a table corner, or a garden bed with string. Ask your child to explain why it works using squares. Safety: help with scissors. This lesson also builds A1 skills: using models to reason and prove.