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

Using the Pythagorean Theorem to Find Unknown Sides

E2.4 A1About 50 minutes

Learning goals

I can find the length of the hypotenuse when I know both legs.
I can find the length of a leg when I know the hypotenuse and the other leg.
I can estimate a square root and use a calculator to find it.
I can solve real problems by finding a right triangle in the situation.

Words to know

Pythagorean theorem
In a right triangle with legs a and b and hypotenuse c, a² + b² = c².

square root (√)
The number that multiplies by itself to give a number: √64 = 8.

perfect square
A whole number that is a whole number times itself, like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

diagonal
A line segment joining two opposite corners of a shape. In a rectangle, it splits the rectangle into two right triangles.

Remember from the last lesson

In a right triangle, a² + b² = c², where c is the hypotenuse (across from the right angle). You used it to test triangles. Now you'll use it to find a side you don't know.

Let's start with a backyard rink

A family floods a backyard rink that is 12 m long and 8 m wide. How long is the diagonal, from one corner to the opposite corner? You can't easily measure it in the snow, but the diagonal makes a right triangle with two sides of the rink. The Pythagorean theorem gives the answer.

A rectangular backyard ice rink with white wooden boards, surrounded by snow at sunset. — The diagonal of a rectangular rink, from one corner to the opposite corner, is the hypotenuse of a right triangle whose legs are the rink's length and width. AI-generated image

Two right triangles in real life, drawn to scale. Left: a 5 m ladder leans against a wall with its foot 1.5 m from the wall; the height h is unknown. Right: a TV screen 120 cm wide and 68 cm tall; the diagonal d is unknown.

Two kinds of problems

Finding the hypotenuse (you know both legs): add the squares, then take the square root.

c = √(a² + b²)

Finding a leg (you know the hypotenuse and one leg): subtract the squares, then take the square root.

a = √(c² − b²)

Check: the hypotenuse must be the longest side. If your leg comes out longer than the hypotenuse, you added when you should have subtracted.

Square roots that aren't whole numbers

Most square roots are not whole numbers. To estimate √208, find the perfect squares on either side: 14² = 196 and 15² = 225. So √208 is between 14 and 15, a bit closer to 14. Then use the √ key on a calculator: √208 ≈ 14.42. Round to a sensible place for the situation (often one decimal place).

Worked example

Example 1: The rink diagonal

Find the diagonal of a rink that is 12 m by 8 m.

The diagonal is the hypotenuse. The legs are 12 m and 8 m.
c² = 12² + 8² = 144 + 64 = 208.
c = √208. Estimate: between 14 and 15. Calculator: 14.42...
Round: about 14.4 m.

The diagonal is about 14.4 m.

Worked example

Example 2: How high does the ladder reach?

A 5 m ladder leans against a wall. Its foot is 1.5 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse (it's across from the right angle between the wall and the ground): c = 5. One leg is 1.5. The height h is the other leg.
h² = 5² − 1.5² = 25 − 2.25 = 22.75.
h = √22.75. Estimate: between 4 and 5 (4² = 16, 5² = 25), close to 5. Calculator: 4.769...
Check: 4.77 is less than 5, so the hypotenuse is still the longest side ✓

The ladder reaches about 4.77 m up the wall.

Worked example

Example 3: A TV screen

TV screens are measured along the diagonal. A screen is 120 cm wide and 68 cm tall. How long is its diagonal?

d² = 120² + 68² = 14 400 + 4 624 = 19 024.
d = √19 024 ≈ 137.9 (137² = 18 769 and 138² = 19 044, so it's just under 138).

The diagonal is about 138 cm.

Worked example

Example 4: Distance on a grid

Find the distance between P(1, 2) and Q(7, 10).

Draw a right triangle with PQ as the hypotenuse. Horizontal leg: 7 − 1 = 6. Vertical leg: 10 − 2 = 8.
PQ² = 6² + 8² = 36 + 64 = 100.
PQ = √100 = 10.

The distance is 10 units.

Example 4. The segment PQ is the hypotenuse of a right triangle with legs 6 (across) and 8 (up), so PQ = √100 = 10 units.

A problem-solving routine

Sketch the situation and find the right angle.
Label the hypotenuse c (across from the right angle) and the legs a and b.
Decide: am I finding the hypotenuse (add) or a leg (subtract)?
Estimate the square root using perfect squares, then calculate.
Check that the hypotenuse is the longest side and the answer makes sense.

Hands-on activity

Measure it two ways

You need: a measuring tape, a calculator, paper and a pencil.

Find a rectangle at home: a table top, a door, a window, a rug or a TV.
Measure its length and width. Use the Pythagorean theorem to predict the diagonal.
Now measure the diagonal. How close was your prediction?
Do it for three different rectangles. Record: length, width, predicted diagonal, measured diagonal.
Bonus: measure a TV screen's width and height (only the screen, not the frame) and calculate its diagonal. TV sizes are often given in inches (1 inch = 2.54 cm). Does your answer match the size on the box or the manual?

What to look for: predictions within about 1 cm of the measurements. Small differences come from measuring, and from corners that are not perfectly square.

Just for fun

A "55-inch TV" is not 55 inches wide. The 55 inches (about 140 cm) is measured along the diagonal. TV makers love the hypotenuse, because it's always the biggest number.

Your turn

A right triangle has legs of 6 cm and 8 cm. Find the hypotenuse.

A right triangle has legs of 5 m and 12 m. Find the hypotenuse.

The hypotenuse is 10 cm and one leg is 6 cm. Find the other leg.

The hypotenuse is 25 m and one leg is 7 m. Find the other leg.

A right triangle has legs of 4 cm and 7 cm. Find the hypotenuse to one decimal place.

A 4 m ladder has its foot 1 m from a wall. How high up the wall does it reach?

Between which two whole numbers is √50?

More practice: warm-up

What is √64?

Legs 3 and 4. What is the hypotenuse?

What is √49?

To find a leg, do you add or subtract the squares?

What is √(9 + 16)?

More practice: on your own

A soccer field is 100 m by 60 m. How long is its diagonal?

A kite string is 50 m long and pulled tight. The kite is directly above a point 30 m away from the person holding it. How high is the kite (above the person's hand)?

Find the distance between (−2, 1) and (4, 9).

A ramp for people who use wheelchairs rises 0.5 m over a horizontal distance of 6 m. How long is the ramp's surface?

Find the diagonal of a square with sides of 10 cm.

Challenge

A TV screen has a diagonal of 140 cm and a width of 122 cm. How tall is the screen?

A box is 30 cm wide, 40 cm long and 120 cm tall. Will a 125 cm hockey stick fit inside, from one bottom corner to the opposite top corner?

An isosceles triangle has two sides of 13 cm and a base of 10 cm. Find its height and its area.

Check your understanding

Quiz

A right triangle has legs of 9 and 12. What is the hypotenuse?

The hypotenuse is 17 and one leg is 8. What is the other leg?

Between which two whole numbers is √40?

A 13 m ladder has its foot 5 m from a wall. How high does it reach?

What is the distance between (0, 0) and (3, 4)?

To find a leg when you know the hypotenuse c and the other leg b, you calculate:

For parents and teachers

Big idea: The Ontario curriculum (E2.4) asks Grade 8 students to apply the Pythagorean theorem to solve problems involving an unknown side length of a right triangle. Your child will add squares to find a hypotenuse and subtract squares to find a leg.

Common difficulties:

Forgetting the square root at the end (giving 225 instead of 15).
Adding when finding a leg, which gives a "leg" longer than the hypotenuse.
Not spotting the right triangle in a word problem. Encourage a quick sketch every time.

How to help: Do the measuring activity together with a table, door or TV. A calculator with a √ key is fine for this lesson, but ask your child to estimate first using perfect squares. Safety: if you talk about ladders at home, only an adult should set one up. This lesson also builds A1 skills: problem solving and checking for reasonableness.