In the given figure, ∠AMN = ∠MBC = 76⁰ If p, q and r are the lengths of AM, MB and BC respectively, then express the length of MN in terms of p, q and r.

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Solution

In △AMN and △ABC
∠AMN = ∠ABC = 76⁰ (Given)
∠A = ∠A (Common)
By AA similarity criterion, △AMN ∼ △ABC
If two triangles are similar, then the ratio of their corresponding sides are proportional
$∴ \frac{AM}{AB} = \frac{MN}{BC}$
$\Rightarrow \frac{AM}{AM + MB} = \frac{MN}{BC} \Rightarrow \frac{a}{a + b} = \frac{MN}{c} \Rightarrow MN = \frac{a c}{a + b}$

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In the given figure, ∠AMN = ∠MBC = 760 If p, q and r are the lengths of AM, MB and BC respectively, then express the length of MN in terms of p, q and r.

In △AMN and △ABC ∠AMN = ∠ABC = 760 (Given) ∠A = ∠A (Common) By AA similarity criterion, △AMN ∼ △ABC If two triangles are similar, then the ratio of their corresponding sides are proportional ∴AMAB=MNBC ⇒AMAM+MB=MNBC⇒aa+b=MNc⇒MN=aca+b

In the given figure, ∠AMN = ∠MBC = 76⁰ If p, q and r are the lengths of AM, MB and BC respectively, then express the length of MN in terms of p, q and r.