E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

The Pythagorean Relationship in Real-Life Situations

E1.5 A1 A2About 70 minutes

Learning goals

I can find an unknown side of a right triangle in a real situation, and decide whether my answer is reasonable.
I can use the converse of the Pythagorean theorem to check whether a corner is square.
I can find diagonals of screens, rooms and boxes, including the space diagonal of a box in two steps.
I can find the distance between two points on a grid or map.

Words to know

hypotenuse
The longest side of a right triangle, opposite the right angle.

legs
The two shorter sides of a right triangle, which form the right angle.

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

converse
The theorem run backwards: if a² + b² = c², then the triangle has a right angle.

Pythagorean triple
Three whole numbers that fit a² + b² = c², such as 3, 4, 5 or 5, 12, 13.

space diagonal
A line through the inside of a box from one corner to the opposite corner.

A hook: will the hockey stick fit?

You are mailing a hockey stick to your cousin in Iqaluit. The stick is 128 cm long, and the longest box you can get is 120 cm. Is it hopeless?

Not if you think in three dimensions. A stick can lie diagonally. Right triangles show up in ladders, ramps, TVs, building corners, maps and boxes, and the Pythagorean theorem finds the missing length every time.

Remember from Grade 8

In a right triangle, a² + b² = c², where c is the hypotenuse. To find the hypotenuse, add the squares and take the square root. To find a leg, subtract: a² = c² − b².

A problem-solving routine

Draw the situation and find the right angle.
Label the hypotenuse (opposite the right angle) and the legs.
Write the equation and solve.
Check: the hypotenuse must be the longest side. Round sensibly and include units.

A 5 m ladder with its foot 1.4 m from the wall. The wall, the ground and the ladder form a right triangle with the ladder as the hypotenuse.

Worked example

Example 1: The ladder

A 5 m ladder leans against a wall with its foot 1.4 m from the wall. How high up the wall does it reach?

The ladder is the hypotenuse; the height h and 1.4 m are the legs.
h² + 1.4² = 5², so h² = 25 − 1.96 = 23.04.
h = √23.04 = 4.8 m.
Check: 4.8 m is less than 5 m. ✓

4.8 m

Worked example

Example 2: TV size

A TV is sold as 55 inches, which is its diagonal. Its width and height are in the ratio 16 : 9. Find its width and height.

Let the width be 16x and the height 9x.
(16x)² + (9x)² = 55², so 256x² + 81x² = 3025 and 337x² = 3025.
x² ≈ 8.976, so x ≈ 2.996.
Width ≈ 16 × 2.996 ≈ 47.9 in; height ≈ 9 × 2.996 ≈ 27.0 in.

About 47.9 in wide and 27.0 in tall

Checking a square corner: the converse

If three sides fit a² + b² = c², the angle opposite c is exactly 90°. Builders use this to check corners of decks, foundations and rooms. Measuring 3, 4 and 5 (or 6, 8, 10) is fast and needs no protractor.

The 3-4-5 check. Builders often use multiples, like 6-8-10 or 9-12-15, for bigger corners.

Worked example

Example 3: Is the deck square?

A deck corner has boards measured 6 ft and 8 ft from the corner. The diagonal between the marks is 10 ft 2 in. Is the corner square?

For a right angle the diagonal should be √(6² + 8²) = √100 = 10 ft.
10 ft 2 in is longer than 10 ft.
A longer diagonal means the angle is a bit more than 90°.

Not square. The boards need to be pushed slightly closer together.

A 120 cm × 40 cm × 30 cm box. The floor diagonal and an upright edge make a right triangle whose hypotenuse is the space diagonal.

Worked example

Example 4: The hockey stick

Will a 128 cm hockey stick fit in a box 120 cm × 40 cm × 30 cm?

Floor diagonal: d² = 120² + 40² = 14 400 + 1600 = 16 000.
Space diagonal: s² = d² + 30² = 16 000 + 900 = 16 900.
s = √16 900 = 130 cm.

Yes. The space diagonal is 130 cm, so a 128 cm stick fits diagonally.

Worked example

Example 5: Shortcut across the park

A rectangular park is 120 m by 50 m. How much shorter is walking along the diagonal than along two sides?

Diagonal: √(120² + 50²) = √(14 400 + 2500) = √16 900 = 130 m.
Along the sides: 120 + 50 = 170 m.
Saving: 170 − 130 = 40 m.

40 m shorter

Worked example

Example 6: Distance on a grid

On a map grid where each square is 1 km, a cabin is at (1, 2) and a lake access point is at (7, 10). How far apart are they in a straight line?

Horizontal change: 7 − 1 = 6 km. Vertical change: 10 − 2 = 8 km.
These are the legs of a right triangle.
d = √(6² + 8²) = √100 = 10 km.

10 km

Watch out!

The hypotenuse is always opposite the right angle and is the longest side.
To find a leg, subtract the squares; don't add.
Square first, then add: √(3² + 4²) = 5, but 3 + 4 = 7. You cannot take square roots of each part separately.
Keep full calculator values until the end, then round.

Hands-on activity

Be the builder

You need: a measuring tape, a pencil, and two walls, a table corner, or a bookshelf.

Choose a corner that should be square (a table, a door frame, a rug, a bed frame).
From the corner, measure and mark 30 cm along one edge and 40 cm along the other.
Measure the diagonal between the marks. If it is 50 cm, the corner is square.
Measure the width and height of a TV, laptop or phone screen. Predict the diagonal with a² + b² = c², then measure it. (Screens are sold by diagonal in inches; divide cm by 2.54 to compare.)

Model answer: a laptop screen 30.5 cm wide and 17.2 cm tall has a diagonal of √(30.5² + 17.2²) ≈ 35.0 cm ≈ 13.8 in.

Just for fun

There are hundreds of different proofs of the Pythagorean theorem. One was published in 1876 by James Garfield, who later became President of the United States.

Your turn

A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.

A right triangle has a hypotenuse of 26 m and one leg of 10 m. Find the other leg.

Is a triangle with sides 7, 24 and 25 a right triangle?

A 6 m ladder has its foot 2 m from a wall. How high does it reach, to the nearest hundredth?

A screen is 30 cm wide and 20 cm tall. Find its diagonal to one decimal.

A box is 60 cm × 25 cm × 80 cm (length × width × height). Find the space diagonal to one decimal.

A kite string is 50 m long and the kite is 30 m away horizontally. How high is the kite (ignore the height of the person)?

More practice: warm-up

Legs 3 and 4. Hypotenuse?

Legs 6 and 8. Hypotenuse?

Legs 5 and 12. Hypotenuse?

Which side of a right triangle is the longest?

A square has sides of 1 m. Its diagonal, to two decimals?

More practice: on your own

A baseball diamond is a square with 90 ft sides. How far is it from home plate to second base (the diagonal), to one decimal?

Two cyclists leave the same spot. One rides 8 km north and the other 15 km east. How far apart are they?

A ramp rises 0.75 m over a horizontal distance of 9 m. How long is the ramp surface, to two decimals?

A carpenter measures a frame: 9 in, 12 in, and a diagonal of 16 in. Is the corner square?

Find the distance between (−2, 3) and (4, 11) on a grid.

Challenge

A 65-inch TV has a 16 : 9 screen. Will it fit in a cabinet opening 150 cm wide?

A room is 4 m × 5 m with a 2.5 m ceiling. What is the longest straight pole that fits inside?

A 24 m radio tower is held by 3 wires from the top to anchors 10 m from its base. Each wire needs 0.5 m extra for knots. How much wire is needed?

Check your understanding

Quiz

A right triangle has legs 8 cm and 15 cm. What is the hypotenuse?

A 10 m ladder has its foot 6 m from a wall. How high does it reach?

Which set of sides makes a right triangle?

A builder checks a corner with 3 ft and 4 ft marks. The diagonal is 4 ft 10 in. The angle is:

How far is it between (0, 0) and (5, 12) on a grid?

To find the space diagonal of a box, you:

For parents and teachers

Big idea: E1.5 asks students to solve problems with the side-length relationship of right triangles in real-life situations, including composite shapes (next lesson). Grade 8 introduced the theorem; Grade 9 applies it in 2D and 3D and uses its converse.

Common difficulties:

Adding when finding a leg. Ask: "Is the side you want the longest one?"
Mislabelling the hypotenuse. It is across from the right angle.
Rounding in the middle of a two-step (space diagonal) problem.

How to help: Do the 30-40-50 corner check together on furniture. If you know anyone in construction, ask them to show how they square a corner; this is the career connection (A2).