In triangle ABC, AC = 3, BC = 4, and AB = 5. A semicircle with its center on segment AC is tangent to AB and BC. Find the radius of the semicircle.
See the following image :
Let the slope of AB = -3/4
And the equation of the line containing AB is y = (-3/4)x + 3
Put this line into standard form and we have that
3x + 4y - 12 = 0
And to find the distance between (0,0) and this line we have that
l 3(0) + 4(y) - 12 l 12 12
________________ = _________ = ___ = 2.4
sqrt (3^2 + 4^2) sqrt (25) 5
Let CE be perpendicular to AB
And let the radius of the semicircle = R
So....we can construcst similar triangles such that
R 2.4
_____ = _____ cross-multiply
3 - R 3
3R = 2.4 ( 3 - R)
3R = 7.2 - 2.4R
5.4R = 7.2
R = 7.2 / 5.4 = 4/3