Question

In the given Fig. PQR is a triangle, right angled at Q. If XY || QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

Answer 1

Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. This is also known as Thales theorem.

Given:

`\angle Q= 90^\circ , XY \ || \ QR, PQ = 6 \ \text{cm}, PY = 4 \ \text{cm} \ \text{and} \ PX : XQ = 1 : 2`

Since,
`XY \ || \ QR`,

`PX/XQ = PY/YR`

[ By Thales theorem (BPT)]

`(1)/(2) = PY/YR`
`[PX : XQ = 1 : 2]`

`(1)/(2) = 4 /(PR - PY)`

`[YR= PR - PY]`

`(1)/(2) = 4 /(PR - 4)`

`PR - 4 = 2 * 4`

`PR - 4 = 8`

`PR = 8 +4`

`PR = 12 \ \text{cm}`

In right
`\Delta PQR`,

`PR^2 = PQ^2 + QR^2`

[ By Pythagoras theorem]

`12^2 = 6^2 + QR^2`
`[\text{Given} : PQ= 6 \ \text{cm}]`

`144 = 36 + QR^2`

`144 - 36 + QR^2`

`108= QR^2`

`QR =√(108) =√(3*36) = 6√(3) \ \text{cm}`

Hence, the lengths of PR and QR is 12 cm and
`6√(3)` cm.