How to find the length of the hypotenuse of a 45/45/90 right isosceles triangle : Pythagorean Theorem - High School Math

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Question

In an isosceles right triangle, two sides equal . Find the length of side .

Answer

This problem represents the definition of the side lengths of an isosceles right triangle. By definition the sides equal , , and $x\sqrt{2}$ . However, if you did not remember this definition one can also find the length of the side using the Pythagorean theorem $(a^{2}+b^{2}=c^{2})$ .

$h^{2}=x^{2}+x^{2}$

$h^{2}=2x^{2}$

$h=\sqrt{2x^{2}}$

$\rightarrow x\sqrt{2}$

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Question

ABCD is a square whose side is units. Find the length of diagonal AC.

Answer

To find the length of the diagonal, given two sides of the square, we can create two equal triangles from the square. The diagonal line splits the right angles of the square in half, creating two triangles with the angles of , , and degrees. This type of triangle is a special right triangle, with the relationship between the side opposite the degree angles serving as x, and the side opposite the degree angle serving as $x\sqrt{2}$ .

Appyling this, if we plug in for we get that the side opposite the right angle (aka the diagonal) is $8\sqrt{2}$

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Question

The area of a square is . Find the length of the diagonal of the square.

Answer

If the area of the square is , we know that each side of the square is , because the area of a square is $s^{2}$ .

Then, the diagonal creates two special right triangles. Knowing that the sides = , we can find that the hypotenuse (aka diagonal) is $12\sqrt{2}$

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