In ΔABC, ∠B = 90o, If the points D and E are on the side BC such that BD = DE = EC, then which of the following is true?
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e 8AE2 = 3AC2 + 5AD2
Explanation:
Let the BD, BE and BC be x, 2x and 3x respectively.
In ΔABD,
(AD)2 = (AB)2 + (BD)2
(AD)2 = (AB)2 + (x)2 -------(1)
In ΔABE,
(AE)2 = (AB)2 + (BE)2
(AE)2 = (AB)2 + (2x)2
(AE)2 = (AB)2 + 4x2 -------(2)
In ΔABC,
(AC)2 = (AB)2 + (BC)2
(AC)2 = (AB)2 + (3x)2
(AC)2 = (AB)2 + 9x2 ---------(3)
Eq(1) multiply by 4,
4AD2 = 4AB2 + 4x2 ------(4)
Eq(4) - (2),
4AD2 - AE2 = 3AB2 -----(5)
Eq(1) multiply by 9,
9AD2 = 9AB2 + 9x2 -----(6)
Eq(6) - (3),
9AD2 - AC2 = 8AB2 -----(7)
Eq(5) multiply by 8 and Eq(7) multiply by 3,
32AD2 - 8AE2 = 24AB2 -----(8)
27AD2 - 3AC2 = 24AB2 -----(9)
From eq(8) and (9),
32AD2 - 8AE2 = 27AD2 - 3AC2
8AE2 = 5AD2 + 3AC2