2 Tangent Semi Circles Tangent in a Ray

[Sophia P]

The figure shows two semicircles with centers P and M. The semicircles ard tangent to each other at point B and ray DE is tangent to both semicircles at points F and E. If PB=BC=6, find DE.

I have gotten most of the question finished, but I can’t find out what FE is, which I think is needed to find DE. Thank you!


Edit: so I’m going to elaborate on what I did so far: I created the two similar triangles, triangle FPD and triangle EMD and was able to put the 6 and 3 values on the shorter legs respectively. I could create a proportion if I knew the value of FE but I don’t know how to get to FE. Thanks!

 

[Harry_the_cat]

The figure shows two semicircles with centers P and M. The semicircles ard tangent to each other at point B and ray DE is tangent to both semicircles at points F and E. If PB=BC=6, find DE.

I have gotten most of the question finished, but I can’t find out what FE is, which I think is needed to find DE. Thank you!

View attachment 10156
Edit: so I’m going to elaborate on what I did so far: I created the two similar triangles, triangle FPD and triangle EMD and was able to put the 6 and 3 values on the shorter legs respectively. I could create a proportion if I knew the value of FE but I don’t know how to get to FE. Thanks!

It will also help if you realise that FP=PB=6 (both radii of the bigger circle), EM=MC=3 (radii of smaller triangle) and that angle PFD = angle MED= 90 degrees (property of tangent to a circle being perpendicular to radius at point of contact).
So triangle PFD is similar to triangle MED.

Let CD =x and then look at FP/EM =PD/MD - solve for x. Then Pythagoras will enable you to calculate DE. Give it a go!

 

[yma16]

if you choose the middle point

The figure shows two semicircles with centers P and M. The semicircles ard tangent to each other at point B and ray DE is tangent to both semicircles at points F and E. If PB=BC=6, find DE.

I have gotten most of the question finished, but I can’t find out what FE is, which I think is needed to find DE. Thank you!

View attachment 10156
Edit: so I’m going to elaborate on what I did so far: I created the two similar triangles, triangle FPD and triangle EMD and was able to put the 6 and 3 values on the shorter legs respectively. I could create a proportion if I knew the value of FE but I don’t know how to get to FE. Thanks!

of FP and name it Z, then triangle PZE is congruent to triangle EMD. So, DB has length 9, and triangle EMD
is a right triangle. Then you can find the length DE.