In the given figure, $\angle\text{AMN}=\angle\text{MBC}=76^\circ.$ 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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In $\triangle\text{ABC}$ and $\triangle\text{AMN},$
$\angle\text{M}=\angle\text{B}=76^\circ$ .....(corresponding angles)
and $\angle\text{BAC}=\angle\text{MAN}$ .....(common angle)
$\Rightarrow\triangle\text{ABC}\sim\triangle\text{AMN}$ .....(AA criterion)
$\frac{\text{AB}}{\text{AM}}=\frac{\text{BC}}{\text{MN}}$
$\frac{\text{AM}+\text{MB}}{\text{AM}}=\frac{\text{BC}}{\text{MN}}$
$\frac{\text{a}+\text{b}}{\text{a}}=\frac{\text{c}}{\text{MN}}$
$\Rightarrow\text{MN}=\frac{\text{ac}}{\text{a}+\text{b}}$
$\angle\text{M}=\angle\text{B}=76^\circ$ .....(corresponding angles)
and $\angle\text{BAC}=\angle\text{MAN}$ .....(common angle)
$\Rightarrow\triangle\text{ABC}\sim\triangle\text{AMN}$ .....(AA criterion)
$\frac{\text{AB}}{\text{AM}}=\frac{\text{BC}}{\text{MN}}$
$\frac{\text{AM}+\text{MB}}{\text{AM}}=\frac{\text{BC}}{\text{MN}}$
$\frac{\text{a}+\text{b}}{\text{a}}=\frac{\text{c}}{\text{MN}}$
$\Rightarrow\text{MN}=\frac{\text{ac}}{\text{a}+\text{b}}$
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