Question 145860: triangleABC is an equilateral triangle,D is a point in BC such that BD is one third of BC.prove 9AD2=7AB2
Answer by vleith(2983)   (Show Source):
You can put this solution on YOUR website! Let the length of the sides be x
Draw in point D. Then label then label BD = x/3. That means DC = 2x/3
Now drop a vertical line from D that intersects AC at E. This forms two right triangles (DEC and ADE).
The angle C is 60 degrees. So triangle DEC has the hypotenuse DC with length 2x/3 and is a 30-60-90 triangle. Use trig to find the lengths of all three sides of DEC (2x/3, x/3 and .866(2x/3) )
Now, since EC is x/3, then AE is 2x/3. DE is ,866(2x/3). ADE is also a right trianlge. You have two sides, so you can use pythagorean theorem to find the third side.
Doing so yields AD^2 = (2x/3)^2 + (0.866(2x/3))^2 = 1.75 (4/9)x^2 = (7/9)x^2
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