Question - Solving for the Lengths of Legs in an Isosceles Right-Angled Triangle
Solution:
This image shows a right-angled triangle with one of the angles being $$45^\circ$$, which makes it an isosceles right-angled triangle (since the other non-right angle must also be $$45^\circ$$). The hypotenuse of this triangle is given as 5 units.In an isosceles right-angled triangle, the lengths of the two legs (sides opposite the $$45^\circ$$ angles) are equal. If we let one of the legs be $$x$$, we can use the Pythagorean theorem to solve for $$x$$:\[x^2 + x^2 = 5^2\]\[2x^2 = 25\]\[x^2 = \frac{25}{2}\]\[x = \sqrt{\frac{25}{2}}\]\[x = \frac{5}{\sqrt{2}}\]Multiplying the numerator and denominator by $$\sqrt{2}$$ to rationalize the denominator, we get:\[x = \frac{5\sqrt{2}}{2}\]So the lengths of the two legs are each $$\frac{5\sqrt{2}}{2}$$ units.
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