Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

We're given a right triangle with sides of length a, b, and c. What is the radius R of the inscribed semicircle shown in the figure?  So here we have two similar triangles. We have the original right triangle and we have this other right triangle.  We know that it's a right triangle because the tangent is always perpendicular to the radius at that point. So this is a right triangle so these guys are similar.  And so that means the ratio of the short leg to the hypotenuse should be constant in both triangles which means that a to c has to be equal to R to b minus R because this whole  length is b and we just remove this R right here. And now we just have to solve for R. So cross multiply to get a b minus a R is c R. Move  over this R and divide by a plus c to get that R is equal to a b over a plus c which is this option right here.

Video Summary

The radius \( R \) of the inscribed semicircle in a right triangle with sides \( a \), \( b \), and \( c \) can be found using the properties of similar triangles. By setting up a proportion between the original triangle and a smaller right triangle formed by the tangent and radius, we find \( \frac{a}{c} = \frac{R}{b-R} \). Solving for \( R \) involves cross-multiplying and simplifying the equation to derive \( R = \frac{ab}{a + c} \). This formula provides the radius of the inscribed semicircle within the right triangle.

Keywords

inscribed semicircle

right triangle

radius formula

similar triangles

proportion