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In ΔABC, BM ⊥ AC and CN ⊥ AB; show that:

ABAC=BMCN=AMAN\dfrac{AB}{AC} = \dfrac{BM}{CN} = \dfrac{AM}{AN}

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Answer

ΔABC is shown in the figure below:

In ΔABC, BM ⊥ AC and CN ⊥ AB; show that: AB/AC = BM/CN = AM/AN. Similarity, Concise Mathematics Solutions ICSE Class 10.

In ΔABM and ΔACN,

∠AMB = ∠ANC [Since, BM ⊥ AC and CN ⊥ AB]

∠BAM = ∠CAN [Common angle]

∴ ∆ABM ~ ∆ACN [By A.A.]

Since corresponding sides of similar triangles are proportional we have,

⇒ ABAC=BMCN=AMAN\dfrac{AB}{AC} = \dfrac{BM}{CN} = \dfrac{AM}{AN}

Hence, proved that ABAC=BMCN=AMAN\dfrac{AB}{AC} = \dfrac{BM}{CN} = \dfrac{AM}{AN}.

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