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# Incircle

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Find the diameter of the incircle for a triangle whose side lengths are 8, 15, and 17.

Apr 14, 2020

#1
+932
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The formula for the radius of an inscribed circle in a triangle is 2 * Area= Perimeter * Radius. In an 8, 15, 17 right triangle, twice the area is 8 * 15= 120 and the perimeter is 8+15+17= 40. So the radius is 120/40=3. The radius is half the diameter so your answer is 3 * 2= 6.

Apr 14, 2020
#2
+109524
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I do not know that formula so here is my method.

The radius is r units.  The encircle centre is the point (r,r)

To find r, the perpendicular distance of the hypotenuse to (r,r) must equal r.

$$r=\frac{|Ax_1+By_1+C|}{\sqrt{A^2+b^2}}\\ r=\frac{|8r+15r-120|}{\sqrt{8^2+15^2}}\\ r=\frac{|23r-120|}{17}\\ r=\frac{120-23r}{17}\\ 17r+23r=120\\ 40r=120\\ r=3$$

Diameter is 6

Apr 14, 2020