Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Sep 13, 2023

Find the lengths of the shorter sides of a 45°-45°-90° triangle with longest side 8. Exact answer.

STEP 1

Assumptions1. The triangle is a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle. . The longest side (hypotenuse) is8.
3. We are looking for the lengths of the other two sides.

STEP 2

In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the sides opposite the $45^{\circ}$ angles are equal in length. Let's denote this length as $a$ . The side opposite the $90^{\circ}$ angle (the hypotenuse) is $\sqrt{2}$ times the length of each of the other sides. We can write this relationship as $Hypotenuse = a \cdot \sqrt{2}$

STEP 3

Now, plug in the given value for the hypotenuse to find the length of the other sides.
$8 = a \cdot \sqrt{2}$

STEP 4

To solve for $a$ , divide both sides of the equation by $\sqrt{2}$ .
$a = \frac{8}{\sqrt{2}}$

STEP 5

Rationalize the denominator by multiplying the numerator and the denominator by $\sqrt{2}$ .
$a = \frac{8 \cdot \sqrt{2}}{2}$

STEP 6

implify the right side of the equation to find the length of the other sides.
$a =4\sqrt{2}$ Both shorter sides of the triangle are $4\sqrt{2}$ units long.

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