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In the adjoining figure, ABC is a triangle in which AB = AC. P is a point on the side BC such that PM ⊥ AB and PN ⊥ AC. Prove that BM × NP = CN × MP.

In the adjoining figure, ABC is a triangle in which AB = AC. P is a point on the side BC such that PM ⊥ AB and PN ⊥ AC. Prove that BM × NP = CN × MP. Similarity, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

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Answer

Consider △ABC

Given, AB = AC

∠ B = ∠ C [Angles opposite to equal sides (Property of isosceles triangle)]

Considering △BMP and △CNP

∠ M = ∠ N = 90°.

∠ B = ∠ C.

So, by AA rule of similarity △BMP ~ △CNP.

As triangles are similar,

⇒BMCN=MPNP⇒BM×NP=CN×MP.\Rightarrow \dfrac{BM}{CN} = \dfrac{MP}{NP} \\[1em] \Rightarrow BM \times NP = CN \times MP.

Hence proved.

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