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In the figure (1) given below, ∠AED = ∠ABC. Find the values of x and y.

In the figure (1) given below, ∠AED = ∠ABC. Find the values of x and y. Similarity, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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Answer

Considering △ABC and △ADE,

∠AED = ∠ABC (Given)

∠A = ∠A (Common angles)

Hence by AA axiom △ABC ~ △ADE.

Since, triangles are similar so ratio of their corresponding sides will be equal.

⇒ADAC=DEBC⇒ADAE+EC=DEBC⇒34+2=y10⇒36=y10⇒y=306⇒y=5.\Rightarrow \dfrac{AD}{AC} = \dfrac{DE}{BC} \\[1em] \Rightarrow \dfrac{AD}{AE + EC} = \dfrac{DE}{BC} \\[1em] \Rightarrow \dfrac{3}{4 + 2} = \dfrac{y}{10} \\[1em] \Rightarrow \dfrac{3}{6} = \dfrac{y}{10} \\[1em] \Rightarrow y = \dfrac{30}{6} \\[1em] \Rightarrow y = 5.

Similarly,

⇒ABAE=BCDE⇒AD+DBAE=BCDE⇒3+x4=10y⇒3+x4=105⇒3+x=405⇒3+x=8⇒x=5.\Rightarrow \dfrac{AB}{AE} = \dfrac{BC}{DE} \\[1em] \Rightarrow \dfrac{AD + DB}{AE} = \dfrac{BC}{DE} \\[1em] \Rightarrow \dfrac{3 + x}{4} = \dfrac{10}{y} \\[1em] \Rightarrow \dfrac{3 + x}{4} = \dfrac{10}{5} \\[1em] \Rightarrow 3 + x = \dfrac{40}{5} \\[1em] \Rightarrow 3 + x = 8 \\[1em] \Rightarrow x = 5.

Hence, the value of x = 5 and y = 5.

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