Question - Calculation of Leg Length in 45-45-90 Triangle
Solution:
This image depicts a right triangle, where one of the angles is 45 degrees and the hypotenuse opposite this angle measures 5 units. Since this is a 45-45-90 triangle, the two legs are congruent. In a 45-45-90 right triangle, the legs are each $$ \frac{1}{\sqrt{2}} $$ (which is the same as $$ \sqrt{2}/2 $$) times the length of the hypotenuse.Let's call the length of each leg $$ x $$. Then:\[ x = \frac{1}{\sqrt{2}} \times 5 = \frac{5}{\sqrt{2}} \]However, it's often preferred to rationalize the denominator, so we multiply the numerator and denominator by $$ \sqrt{2} $$:\[ x = \frac{5\sqrt{2}}{\sqrt{2}\times\sqrt{2}} = \frac{5\sqrt{2}}{2} \]Therefore, the length of each leg of the triangle is $$ \frac{5\sqrt{2}}{2} $$ units.
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