Question

If $\Delta ABC\sim \Delta PQC$, then choose the correct options (All lengths are in cm).

• A. $\angle A=\angle P={45}^{\circ }$
• B. $\angle A=\angle P={60}^{\circ }$
• C. $PQ=1cm$
• D. $PQ=2cm$

Answer 1

Answer: (A), (C)

The correct options are
A $\angle A=\angle P={45}^{\circ }$
C $PQ=1cm$

If two triangles are similar, then corresponding angles are equal and corresponding sides are proportional.
$\Delta ABC\sim \Delta PQC\Rightarrow \angle A=\angle P\Rightarrow \angle B=\angle Q\Rightarrow \angle C=\angle C$
$\angle A+\angle B+\angle C={180}^{\circ }\left[Anglesumproperty\right]\angle A+{90}^{\circ }+{45}^{\circ }={180}^{\circ }\angle A={180}^{\circ }-{135}^{\circ }\angle A={45}^{\circ }\angle A=\angle P={45}^{\circ }$
Corresponding sides are proportional.
$\Rightarrow \frac{AB}{PQ}=\frac{BC}{QC}=\frac{AC}{PC}\Rightarrow \frac{3}{PQ}=\frac{2+1}{1}\Rightarrow \frac{3}{PQ}=\frac{3}{1}\Rightarrow PQ=1PQ=1cm$