SOLUTION: A triangle has two 13-in sides and a 10-in side. The largest circle that fits inside this triangle meets each side at a point of tangency. These points of tangency divide the sides

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Question 629134: A triangle has two 13-in sides and a 10-in side. The largest circle that fits inside this triangle meets each side at a point of tangency. These points of tangency divide the sides of the triangle into segments of what lengths? What is the radius of the circle?
Answer by Charles3475(23) (Show Source):
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The triangle is isosceles. Draw the triangle with the 10 inch side as the base and the 13 inch sides on either side. Then draw the circle inside the triangle (the inscribed circle) such that it touches the triangle at the three points of tangency. A vertical line through the apex of the triangle divides the figure into two symmetrical halves and forms a right angle with the 10 inch base. Thus by symmetry, we see the 10 inch base is bisected into two equal segments of 5 inches at the point of tangency (where the circle touches the triangle).

The line that bisects the isosceles triangle (and the figure) from the apex of the triangle to the base forms a right angle with the base. Using the Pythagorean theorem we can compute the length of this line segment which is a leg of a right triangle with a 13 inch hypotenuse and a leg of 5 inches The line segment is computed to be 12 inches.

Next draw lines from the points of tangency to the center of the circle. These lines are perpendicular to the sides of the triangle as lines tangent to the side of a circle are are perpendicular to the radius from the center of the circle to the point of tangency. Now draw a line from either of the base vertices to the center of the circle. This line forms the hypotenuse for two adjacent right triangles. Examination of the two adjacent right triangles shows they are congruent as both have a side the is the radius of the triangle, both share the same hypotenuse, and both have a right angle. As a result the triangles are congruent (by Hypotenuse Leg) and the third sides have equivalent lengths which we know to be 5 inches.

So the 13 inch sides of the triangle are divided into a 5 inch segment and an 8 inch segment.

We may then compute the base angles as the inverse tangent or arctan(12/5) = 67.38 degrees. The base angle is divided in half by the line segment from the base vertex to the center of the circle that formed the two congruent right triangles discussed previously. Using this smaller angle (half of 67.38 degrees) and the 5 inch side of the right triangle we know that tan(67.38/2) = r/5 where r is the radius of the inscribed circle. The radius is computed as 3.33 inches