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In right triangles, there is a fundamental relationship between the lengths of the *legs* and the length of the hypotenuse. This relationship is presented in the Pythagorean Theorem.

A **right triangle** is a specific type of triangle that contains a right angle. Its sides are commonly called the legs

and the hypotenuse.

For a right triangle with legs $a$ and $b,$ and hypotenuse $c,$ the following is true.

$a^2+b^2=c^2$

This can be proven using area.

Draw four copies of one right triangle with legs $a$ and $b$ and hypotenuse $c.$ Arrange them so they create a square.

$(a+b)^2=2ab+c^2$

ExpandPosPerfectSquare$(a+b)^2=a^2+2ab+b^2$

$a^2+2ab+b^2=2ab+c^2$

SubEqn$\text{LHS}-2ab=\text{RHS}-2ab$

$a^2+b^2=c^2$

Find the length of $x.$

Show Solution

Since the triangle is a right triangle, we can use the Pythagorean Theorem to find the length of $x.$

The hypotenuse is the side opposite the right angle, for this triangle it measures $20$ units. The two other sides are the legs, $x$ and $12.$ We can now form an equation with the Pythagorean Theorem, that can be solved for $x$.$a^2+b^2=c^2$

SubstituteValuesSubstitute values

$12^2+x^2=20^2$

CalcPowCalculate power

$144+x^2=400$

SubEqn$\text{LHS}-144=\text{RHS}-144$

$x^2=400-144$

SubTermSubtract term

$x^2=256$

SqrtEqn$\sqrt{\text{LHS}}=\sqrt{\text{RHS}}$

$x=\pm\sqrt{256}$

CalcRootCalculate root

$x=\pm16$

$x \gt 0$

$x=16$

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