Geometry Practice Test 1

D and E are two points of AB and AC.
DE | | BC
AD = 6
BD = 12 - x
AE = 2x
EC = 16 - 2x

In ΔABC,
$\frac{\mathit{A}\mathit{D}}{\mathit{A}\mathit{B}}$ = $\frac{\mathit{A}\mathit{E}}{\mathit{A}\mathit{C}}$ ⇒ $\frac{\mathit{A}\mathit{D}}{\mathit{A}\mathit{D}+\mathit{D}\mathit{B}}$ = $\frac{\mathit{A}\mathit{E}}{\mathit{A}\mathit{E}+\mathit{E}\mathit{C}}$
⇒ $\frac{6}{6+12-\mathit{x}}$ = $\frac{2\mathit{x}}{2\mathit{x}+16-2\mathit{x}}$
⇒ $\frac{6}{18-\mathit{x}}$ = $\frac{2\mathit{x}}{16}$
96 = $36\mathit{x}-2{\mathit{x}}^2$
= $2{\mathit{x}}^2$ − 36x + 96 = 0
${\mathit{x}}^2$ −18x + 32 = 0
${\mathit{x}}^2$ −16x − 2x + 32 = 0
x(x −16)− 2(x −16) = 0
x = 16, x = 2
x = 16 → impossible
So, x = 2 cm