If Δ PQR∼ ΔXYZ, QR = 3 cm, YZ = 4 cm, ar ΔPQR= 54 cm^2, then ar ΔXYZ = ……….. (A) 13.5 cm^2

Correct option is (C) 96 cm²

We have \(\triangle PQR\sim\triangle XYZ,\)

QR = 3 cm, YZ = 4 cm and \(ar(\triangle PQR)=54\,cm^2\)

We know that the area of two similar triangles are in the ratio of the squares of their corresponding sides.

\(\because\) \(\triangle XYZ\sim\triangle PQR\)

\(\therefore\) \(\frac{ar(\triangle XYZ)}{ar(\triangle PQR)}=(\frac{XY}{PQ})^2\)

\(=(\frac{YZ}{QR})^2=(\frac{XZ}{PR})^2\)

\(\Rightarrow\) \(\frac{ar(\triangle XYZ)}{ar(\triangle PQR)}=(\frac{YZ}{QR})^2=\frac{YZ^2}{QR^2}\)

\(=\frac{4^2}{3^2}=\frac{16}{9}\)

\(\Rightarrow\) \(ar(\triangle XYZ)=\frac{16}9\,ar(\triangle PQR)\)

\(=\frac{16}9\times54\)       \((\because ar(\triangle PQR)=54\,cm^2)\)

\(=16\times6\) \(=96\,cm^2\)

Correct option is: (C) 96 cm²