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In the given figure, AB ⟂ BC and DE ⟂ BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD.

In the given figure, AB ⟂ BC and DE ⟂ BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD. Similarity of Triangles, RSA Mathematics Solutions ICSE Class 10.

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Answer

Given,

AB ⟂ BC and DE ⟂ BC

Thus AB ∥ DE.

∠ABC = ∠DEC [Given]

∠ACB = ∠DCE [Common angle in both triangles]

∴ ΔABC ∼ ΔDEC (By A.A. axiom)

From figure,

DC = AC - AD = 24 - AD

We know that,

Corresponding sides of similar triangles are proportional.

∴DEAB=DCAC⇒39=24−AD24⇒24−AD=3×249⇒24−AD=729⇒24−AD=8⇒AD=24−8⇒AD=16 cm.\therefore \dfrac{DE}{AB} = \dfrac{DC}{AC} \\[1em] \Rightarrow \dfrac{3}{9} = \dfrac{24 - AD}{24} \\[1em] \Rightarrow 24 - AD = \dfrac{3 \times 24}{9} \\[1em] \Rightarrow 24 - AD = \dfrac{72}{9} \\[1em] \Rightarrow 24 - AD = 8 \\[1em] \Rightarrow AD = 24 - 8 \\[1em] \Rightarrow AD = 16 \text{ cm.}

Hence, AD = 16 cm.

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