Question - Solving for Leg Lengths in a 45-45-90 Triangle
Solution:
The image shows a right triangle with one angle of 45 degrees and the hypotenuse having a length of 3 units. Given that the triangle is a right triangle with a 45-degree angle, we can infer that this is a 45-45-90 triangle, which is an isosceles right triangle. In such triangles, the lengths of the legs are equal, and the length of the hypotenuse is √2 times the length of each leg.Let's use the properties of a 45-45-90 triangle to find the lengths of the two legs. If $$ a $$ is the length of one leg, we have:$$ a \cdot \sqrt{2} = 3 $$Now we solve for $$ a $$:$$ a = \frac{3}{\sqrt{2}} $$To rationalize the denominator, we multiply the numerator and the denominator by $$ \sqrt{2} $$:$$ a = \frac{3}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} $$$$ a = \frac{3 \cdot \sqrt{2}}{2} $$So both legs of the triangle are $$ \frac{3 \cdot \sqrt{2}}{2} $$ units long.
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