Q.5. In the figure(9.67), AB||CD||EF||GH and AX = XY = YH. If AC = 1.5 cm find AG.

Ans=4.5cm
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In the figure in ab|| cd||ef||gh and ad=xy=yh if as= 1.5 cm find ag
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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} AG AC ​ = AH AX ​ \implies \frac{AC}{AG} = \frac{AX}{AX\: +\: XY \:+\: YH}⟹ AG AC ​ = AX+XY+YH AX ​ ∵ AX = XY = YH (given) \implies \frac{AC}{AG} = \frac{AX}{AX\: +\: AX \:+\: AX}⟹ AG AC ​ = AX+AX+AX AX ​ \implies \frac{AC}{AG} = \frac{AX}{3AX}⟹ AG AC ​ = 3AX AX ​ \implies \frac{AC}{AG} = \frac{1}{3}⟹ AG AC ​ = 3 1 ​ substituting AC = 1.5 cm \implies \frac{1.5}{AG} = \frac{1}{3}⟹ AG 1.5 ​ = 3 1 ​ on cross-multiplication \implies AG = 1.5 imes 3⟹AG=1.5×3 \implies \bold{AG = 4.5\:cm}⟹AG=4.5cm Thus, \boxed{\bold{\underline{AG = 4.5\:cm}}} AG=4.5cm ​ ​ .
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