Perpendicular from Right Angle to Hypotenuse Divides the Triangle into Two Similar Triangles
ABC is an equ...
Question
ABC is an equilateral triangle. If D is a point on side BC such that BD : BC =1:3, then AD2:AB2 = .
A
7 : 9
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B
4 : 3
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C
11 : 9
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D
17 : 3
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Solution
The correct option is A 7 : 9
Given: ABC is an equilateral triangle
BD = 13BC = 13AB
In △ABC, BE=12BC=12AB AE2=AB2−BE2 ⇒AE2=AB2−(12AB)2 ⇒AE2=AB2−14AB2 ⇒AE2=34AB2 ---(1)
In △ADE, AE2=AD2−DE2 ⇒AE2=AD2−(BE−BD)2 ⇒AE2=AD2−(12AB−13AB)2 ⇒AE2=AD2−(16AB)2 ⇒AE2=AD2−136AB2 ---(2)
From (1) and (2), 34AB2 = AD2−136AB2 ⇒AD2=34AB2+136AB2 ⇒AD2=AB2×(27+136) ⇒AD2=2836AB2 ⇒AD2=79AB2 AD2:AB2 = 7 : 9