Question

AB||CD|EF||GH and AX = XY = YH. If AC = 1.5 cm, find AG.

Answer 1

Given:

AB // CD // EF // GH

AX = XY = YH

AC = 1.5 cm

To find:

The value of AG

Solution:

Since CD // GH .... (given)

∴ CX // GH ... (from the given figure)

Considering Δ ACX and Δ AGH, we get

∠CAX = ∠GAH ..... [common angle]

∠CAX = ∠AGH ...... [corresponding angles]

∴ Δ ACX ~ Δ AGH .... [By AA similarity]

Now, we know that the corresponding sides of two similar triangles are proportional to each other.

So, from similar Δ ACX & Δ AGH, we get

∴ $\frac{AC}{AG} = \frac{AX}{AH}$

$\implies \frac{AC}{AG} = \frac{AX}{AX\: +\: XY \:+\: YH}$

∵ AX = XY = YH (given)

$\implies \frac{AC}{AG} = \frac{AX}{AX\: +\: AX \:+\: AX}$

$\implies \frac{AC}{AG} = \frac{AX}{3AX}$

$\implies \frac{AC}{AG} = \frac{1}{3}$

substituting AC = 1.5 cm

$\implies \frac{1.5}{AG} = \frac{1}{3}$

on cross-multiplication

$\implies AG = 1.5 \times 3$

$\implies \bold{AG = 4.5\:cm}$

Thus, $\boxed{\bold{\underline{AG = 4.5\:cm}}}$.

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