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# In triangle JKL, we have JK = JL = 25 and KL = 30. Find the inradius of triangle JKL and the circumradius of triangle JKL.

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In triangle JKL, we have JK = JL = 25 and KL = 30. Find the inradius of triangle JKL and the circumradius of triangle JKL.

Jul 2, 2022

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Jul 2, 2022
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Draw altitude $$JM$$. Because the triangle is isosceles, $$KM = ML = 15$$

Then, by the Pythagorean Theorem, $$JM = \sqrt{25^2 - 15^2} = 20$$.

This means that the area of the triangle is $$2 \times( 20 \times 15 \div 2) = 300$$.

Now, recall the formula for inradius, which is $${\text{Area} \over \text{semiperimeter} } = {300 \over 40} = \color{brown}\boxed{7.5}$$

Now, recall the formula for circumradius, which is $${\text{product of the side lengths} \over 4\times \text{Area}} = {18,750 \over 1,200} = \color{brown}\boxed{15.625}$$

Jul 2, 2022