Problem:
Triangle ABC has a right angle at C,AC=3 and BC=4. Triangle ABD has a right angle at A and AD=12. Points C and D are on opposite sides of AB. The line through D parallel to AC meets CB extended at E. If DBDE=nm, where m and n are relatively prime positive integers, then m+n=
Answer Choices:
A. 25
B. 128
C. 153
D. 243
E. 256
Solution:
Since AB=32+42=5 and BD=52+122=13, it follows that
Draw AG parallel to CE with G on DE. Then ∠GAD=∠CAB since both are complementary to ∠GAB. Thus, triangles GAD and CAB are similar, and GD=ABADCB=548. Hence DE=563. Apply the Pythagorean theorem to triangles ABC and DAB to find DB=13. Therefore, DBDE=1363/5=6563, so m+n=128.
