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Side lengths

Pythagorean theorem

Japanese school year: Junior high 3

What you learn

Find side lengths of right triangles using integers and square roots.

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Key points

This is the Pythagorean theorem showing the relationship between the three sides of a right triangle. The sum of the squares of the two legs aa and bb equals the square of the hypotenuse cc.

a2+b2=c2a^2 + b^2 = c^2

This formula calculates the hypotenuse cc of a right triangle directly. Add the squares of the two shorter sides aa and bb, then take the square root.

c=a2+b2c = \sqrt{a^2 + b^2}

Use this ratio to find side lengths in a 45∘45^\circ-45∘45^\circ-90∘90^\circ isosceles right triangle. When each of the two equal legs is 1, the hypotenuse is 2\sqrt{2}.

1:1:21 : 1 : \sqrt{2}

Use this ratio to find side lengths in a 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle. The shortest leg is 1, the other leg is 3\sqrt{3}, and the hypotenuse is 2.

1:3:21 : \sqrt{3} : 2
Worked example

In the right triangle shown, find the length x.

Right triangle ABC. Angle B is a right angle. AB is 3 cm, BC is 5 cm, and CA is x.ABC3 cm5 cmx
By the Pythagorean theorem, x2=52+32=34,x=34 cmx^2 = 5^2 + 3^2 = 34,\quad x = \sqrt{34}\,\mathrm{cm}

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