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In the adjoining figure, DE || BC and all measurements are in centimeters. The length of AE is

In the adjoining figure, DE || BC and all measurements are in centimeters. The length of AE is (a) 2 cm (b) 2.25 cm (c) 3.5 cm (d) 4 cm. Similarity, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.
  1. 2 cm

  2. 2.25 cm

  3. 3.5 cm

  4. 4 cm

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Answer

Given DE || BC.

Considering △ABC and △ADE,

∠ A = ∠ A (Common angles)
∠ ABC = ∠ ADE (Corresponding angles are equal)

Hence, by AA axiom △ABC ~ △ADE.

Let the length of AE be x cm.

Since, both the triangles are similar,

ADAB=AEACADAD+DB=AEAE+EC33+4=xx+337=xx+33(x+3)=7x3x+9=7x4x=9x=2.25\therefore \dfrac{AD}{AB} = \dfrac{AE}{AC} \\[1em] \Rightarrow \dfrac{AD}{AD + DB} = \dfrac{AE}{AE + EC} \\[1em] \Rightarrow \dfrac{3}{3 + 4} = \dfrac{x}{x + 3} \\[1em] \Rightarrow \dfrac{3}{7} = \dfrac{x}{x + 3} \\[1em] \Rightarrow 3(x + 3) = 7x \\[1em] \Rightarrow 3x + 9 = 7x \\[1em] \Rightarrow 4x = 9 \\[1em] \Rightarrow x = 2.25

∴ Length of AE = 2.25 cm.

Hence, Option 2 is the correct option.

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