Problem: Given right triangle ABC with legs 3 and 4. Find the length of the angle trisector to the hypotenuse nearer the shorter leg.
Answer Choices:
A. 13323−24
B. 13123−9
C. 63−8
D. 6510
E. 1225
Solution:
Since CD trisects right angle C,∠BCD=30∘ and ∠DCA=60∘ and ∠CDE=30∘, so that △DEC is a 30∘−60∘−90∘ triangle. Let EC=x
∴DE=x3 and DC=2x
x34−x=34,x=3+4312,2x=3+4324=13323−24
or
Area of △BCD=21⋅3⋅2x⋅sin30∘=23x
Area of △ACD=21⋅4⋅2x⋅sin60∘=23x
Area of △BCD+ Area of △ACD= Area of △ACB
∴23x+23x=6,x=3+4312=13163−12,
2x=13323−24
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