Problem:
Triangle ABC is a right triangle with ∠ACB as its right angle, m∠ABC=60∘, and AB=10. Let P be randomly chosen inside △ABC, and extend BP to meet AC at D. What is the probability that BD>52​ ?
Answer Choices:
A. 22−2​​
B. 31​
C. 33−3​​
D. 21​
E. 55−5​​ Solution:
Since AB is 10 , we have BC=5 and AC=53​. Choose E on AC so that CE=5. Then BE=52​. For BD to be greater than 52​,P has to be inside △ABE. The probability that P is inside △ABE is
Area of △ABC Area of △ABE​=21​CA⋅BC21​EA⋅BC​=ACEA​=53​53​−5​=3​3​−1​=33−3​​​.
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