Question

(a) If $△ A BC \sim △ PQR$ in which $A B = 6$ cm, $BC = 4$ cm, $A C = 8$ cm and $PR = 6$ cm, then find the length of $(PQ + QR)$ .

(b) In the given figure, $\frac{QR}{QS} = \frac{QT}{PR}$ and $∠1 = ∠2$ , show that $△ PQS \sim △ TQR$ .

Approach

Given $△ A BC \sim △ PQR$ with $A B = 6, BC = 4, A C = 8, PR = 6$ , find $PQ + QR$ . Use similarity ratio $\frac{A B}{PQ} = \frac{BC}{QR} = \frac{A C}{PR}$ . From $\frac{8}{6} = \frac{4}{3}$ , compute $PQ = 4.5, QR = 3$ . Sum is $7.5$ cm.

Step-by-step working

1
Similarity ratio: $\frac{A B}{PQ} = \frac{BC}{QR} = \frac{A C}{PR} = \frac{6}{PQ} = \frac{4}{QR} = \frac{8}{6}$
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2
$\frac{6}{PQ} = \frac{8}{6} \Rightarrow PQ = \frac{36}{8} = 4.5$ cm
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3
$\frac{4}{QR} = \frac{8}{6} \Rightarrow QR = \frac{24}{8} = 3$ cm
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4
$PQ + QR = 4.5 + 3 = 7.5$ cm
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Concepts used

similar trianglesratio of sidesfind lengthprove similarity

Approach

Step-by-step working

Concepts used

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