If the $Δ A B C \sim Δ P Q R,$ perimeter of $Δ A B C = 32 c m,$ perimeter of $Δ P Q R = 48 c m$ and $P R = 6 c m,$ then find the length of $A C .$

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Solution

Since $Δ A B C \sim Δ P Q R$ we have
$\frac{A B}{P Q} = \frac{B C}{Q R} = \frac{A C}{P R}$
Here we will invoke a property of ratio and proportion; if we have
$\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}} = . . . . . . = \frac{a_{n}}{b_{n}}$
then we will have
$\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{a_{3}}{b_{3}} = . . . . . . = \frac{a_{n}}{b_{n}} = \frac{a_{1} + a_{2} + a_{3} + . . . . . . a_{n}}{b_{1} + b_{2} + b_{2} + . . . . . . b_{n}}$
So we can write,
$\frac{A B}{P Q} = \frac{B C}{Q R} = \frac{A C}{P R} = \frac{A B + B C + A C}{P R + Q R + P R}$
$∴$ $\frac{A C}{P R} = \frac{p e r i m e t e r o f Δ A B C}{p e r i m e t e r o f Δ P Q R}$
$\Rightarrow \frac{A C}{6} = \frac{32}{48}$
$\Rightarrow A C = \frac{32}{8} = 4 c m$

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If the ΔABC∼ΔPQR, perimeter of ΔABC=32cm, perimeter of ΔPQR=48cm and PR=6cm, then find the length of AC.

If the $Δ A B C \sim Δ P Q R,$ perimeter of $Δ A B C = 32 c m,$ perimeter of $Δ P Q R = 48 c m$ and $P R = 6 c m,$ then find the length of $A C .$