In the given figure, if ∠𝐴 = 90°, ∠𝐵 = 90°, 𝑂𝐵 = 4. 5 𝑐𝑚, 𝑂𝐴 = 6 𝑐𝑚 and 𝐴𝑃 = 4 𝑐𝑚, then find 𝑄𝐵.

We are given the following measurements of the figure, ∠𝐴 = 90°, ∠𝐵 = 90°, 𝑂𝐵 = 4. 5 𝑐𝑚, 𝑂𝐴 = 6 𝑐𝑚 and 𝐴𝑃 = 4 𝑐𝑚 and we need to find 𝑄𝐵.

In ∆𝑂𝐴𝑃 and ∆𝑂𝐵𝑄, ∠𝐴 = ∠𝐵 = 90° (given)

Also, ∠𝑃𝑂𝐴 and ∠𝑄𝑂𝐵 are vertically opposite angles.

∠𝑃𝑂𝐴 = ∠𝑄𝑂𝐵 ⇒∆𝑂𝐴𝑃~∆𝑂𝐵𝑄 by 𝐴𝑆𝐴 (angle-side-angle) criterion.

Therefore, $\frac{OA}{OB}=\frac{PA}{QB}$

$\begin{array}{r}\text{ Now, }\frac{OA}{OB}=\frac{6}{4.5}\\ \Rightarrow \frac{OA}{OB}=\frac{4}{3}\\ \frac{PA}{QB}=\frac{4}{BQ}\\ \Rightarrow \frac{6}{4.5}=\frac{4}{BQ}\\ \Rightarrow QB=3\mathrm{cm}\end{array}$

Hence, the length of the side 𝑄𝐵 is 3 𝑐𝑚.