MTH1W The Pythagorean Theorem: Notes and Practice

A rectangular park is 120 m long and 50 m wide. Walking along two edges from one corner to the opposite corner is $120 + 50 = 170$ m. Cutting straight across on the diagonal path is shorter, but by how much? You cannot measure the path with the edges alone, unless you know one of the most useful facts in all of mathematics. The same fact tells you how tall a ladder can reach, what "a 55-inch TV" really measures, and lets a builder check that the corner of a deck is perfectly square using only a tape measure. It is called the Pythagorean theorem, and it connects the three sides of every right triangle.

Legs and hypotenuse

A right triangle has one $9 0^{\circ}$ angle. The two sides that form the right angle are the legs, usually called $a$ and $b$ . The side across from the right angle is the hypotenuse, called $c$ . The hypotenuse is always the longest side of a right triangle, because it is opposite the largest angle. The theorem says: in a right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs.

a^{2} + b^{2} = c^{2}

Fig 1A right triangle with legs 3 and 4 and hypotenuse 5, drawn to scale, with a square on each side divided into 9, 16 and 25 unit squares

Why it works. Take four copies of a right triangle with legs $a$ and $b$ , and arrange them inside a big square with side $a + b$ , one triangle in each corner, so that their hypotenuses form a tilted square of side $c$ in the middle. The big square's area can be counted two ways: as $(a + b)^{2}$ , or as the tilted square plus the four triangles, $c^{2} + 4 \times \frac{1}{2} ab$ . Expanding and setting the two equal:

a^{2} + 2 ab + b^{2} = c^{2} + 2 ab \Rightarrow a^{2} + b^{2} = c^{2}

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In right triangle

PQR

, the right angle is at

Q

. Which side is the hypotenuse?

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The squares on the two legs of a right triangle have areas

36 cm^{2}

and

64 cm^{2}

. What is the area of the square on the hypotenuse?

cm²

Finding a missing side

You know	You want	Formula	Example
Both legs a and b	The hypotenuse c	$c = a^{2} + b^{2}$	$a = 5, b = 12$ : $c = 25 + 144 = 169 = 13$
The hypotenuse c and one leg b	The other leg a	$a = c^{2} - b^{2}$	$c = 10, b = 6$ : $a = 100 - 36 = 64 = 8$

Add the squares for the hypotenuse, subtract for a leg

Two checks before you calculate

First, the theorem is only for right triangles; for any other triangle,

a^{2} + b^{2}

does not equal

c^{2}

. Second, decide which side is the hypotenuse. If you are finding the hypotenuse, add the squares. If you are finding a leg, subtract the smaller square from the hypotenuse's square. Your answer for a leg must come out shorter than the hypotenuse.

Fig 2Left: a 120 m by 50 m park drawn to scale, whose diagonal is the hypotenuse, 130 m, found by adding the squares. Right: a 10 m ladder with its foot 6 m from a wall, which reaches 8 m up the wall, a leg found by subtracting the squares

Many answers are not whole numbers. With legs 7 cm and 10 cm, $c = 49 + 100 = 149$ . The value $149$ is the exact answer; $12.2$ cm is that value rounded to one decimal place. Round only at the very end, and say how you rounded. Real problems often hide a right triangle inside a bigger shape (a composite shape): the height of a triangular roof, the diagonal brace of a gate, the slanted edge of a ramp, or the straight-line distance between two points on a grid. Sketch the shape, find the right angle, and label the legs and hypotenuse before you calculate.

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A right triangle has a hypotenuse of 20 cm and one leg of 12 cm. How long is the other leg?

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A right triangle has legs 4 cm and 6 cm. How long is the hypotenuse, to the nearest tenth of a centimetre?

VideoAn Ontario teacher uses the Pythagorean theorem to find a missing side of a right triangle.Watch on YouTube

The converse: checking for a right angle

The theorem also works backwards. If the sides of a triangle satisfy $a^{2} + b^{2} = c^{2}$ , where $c$ is the longest side, then the triangle is a right triangle. Builders use this to check corners: measure 3 units along one wall and 4 units along the other; if the diagonal between those marks is exactly 5 units, the corner is $9 0^{\circ}$ . (Bigger multiples like 6, 8, 10 feet or 0.9, 1.2, 1.5 m give a more precise check.) If $a^{2} + b^{2}$ is less than $c^{2}$ , the longest side is too long for a right angle, so the angle opposite it is obtuse. If $a^{2} + b^{2}$ is greater than $c^{2}$ , that angle is acute, and since it is the largest angle, the whole triangle is acute.

Fig 3Three triangles drawn to scale with sides 3 and 4 and a third side of 4.5, 5 or 6: since 25 is more than, equal to or less than the third side squared, the angle opposite it is acute (about 79 degrees), right (90 degrees) or obtuse (about 117 degrees)

Pythagorean triple	Check	A multiple of it
3, 4, 5	9 + 16 = 25	6, 8, 10
5, 12, 13	25 + 144 = 169	10, 24, 26
8, 15, 17	64 + 225 = 289	16, 30, 34
7, 24, 25	49 + 576 = 625	14, 48, 50

Whole-number right triangles worth recognizing

An old idea from many places

The theorem is named after Pythagoras, a Greek thinker of the 500s BCE, but the relationship was known long before him and in many places. A Babylonian clay tablet known as Plimpton 322, about 3800 years old (dates vary), lists sets of numbers that fit right triangles. Early texts from India (the Sulba Sutras) and China (where it is known as the gougu rule) also describe it; their exact dates are uncertain. Historians are not sure whether Pythagoras himself proved it.

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A triangle has sides 6, 7 and 9. What kind of triangle is it?

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A builder marks 0.9 m along one wall and 1.2 m along the other. For a square corner, how long should the diagonal between the marks be?

A broken clay tablet covered in rows and columns of small wedge-shaped marks. — PhotoPlimpton 322 is a Babylonian clay tablet about 3800 years old. Scholars link its columns of numbers to the sides of right triangles, recorded long before Pythagoras.photo author unknown, Public domain, via Wikimedia Commons

The Pythagorean Theorem

Key things to remember

Legs and hypotenuse

Finding a missing side

The converse: checking for a right angle

Worked examples

Where marks get lost

Flashcards 8

Practice