In the figure, given below, PQR is a right-angled triangle at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

Given XY || QR,

By basic proportionality theorem :

$\therefore \dfrac{PX}{XQ} = \dfrac{PY}{YR} \\[1em] \Rightarrow \dfrac{1}{2} = \dfrac{4}{YR} \\[1em] \Rightarrow YR = 4 \times 2 = 8 \text{ cm}.$

From figure,

PR = PY + YR = 4 + 8 = 12 cm.

Since, PQR is a right-angled triangle.

By pythagoras theorem we get,

$\Rightarrow PR^2 = PQ^2 + QR^2 \\[1em] \Rightarrow 12^2 = 6^2 + QR^2 \\[1em] \Rightarrow 144 = 36 + QR^2 \\[1em] \Rightarrow QR^2 = 144 - 36 \\[1em] \Rightarrow QR^2 = 108 \\[1em] \Rightarrow QR = \sqrt{108} = 10.392 \text{ cm}.$

Hence, PR = 12 cm and QR = 10.392 cm.