The measure of m, representing the length of segment HY along the angle bisector HU in triangle GHY, is 20/3.
In triangle GHY, U is the incenter, and we are tasked with finding the measure of m, where m is the length of the segment HY along the angle bisector HU. The property of the incenter states that the incenter divides the angle bisectors of a triangle into segments that are proportional to the sides of the triangle.
Let HU be denoted as 'a', MU as 'b', and HY as 'm'. According to the property, we have:
HU / MU = GH / HY
Given that HU = 21, MU = 5, and GH = 28, we can solve for HY:
21 / 5 = 28 / m
Cross-multiplying, we get:
21m = 5 * 28
21m = 140
m = 140 / 21
Simplifying, we find:
m = 20 * 7 / (3 * 7)
m = 20 / 3
Therefore, the measure of m, the length of HY, is 20/3.