Question

Triangle DEF is circumscribed about circle R. Points S, T and U are points of tangency where SD = 4 m and UF = 7 m. What is the measure of DF

Answer 1

The measure off DF is 11.

In a circle inscribed within a triangle, the distance from each vertex of the triangle to the two nearest touchpoints (points of tangency on the circle) are equal. Since SD=4, DT=4 as well. Since UF=7, then FT=7.

DF=DT+TF=4+7=11.

Answer 2

First of all, we need to establish some known things. Let us call the radius r and the center of the circle R. Consider RD and then consider the 2 triangles DUR and DSR. We have that they have both a 90 degree angle (due to the tangency condition), they have DR common and also SR is equal to UR since both are equal to r. Hence, we have that the triangles are equal, thus, we have that DS=DU. Since DF=DU+UF, we have that DF=11