Question

In the adjoining figure, ∠PQR = 90° and XY || QR. If PX : QX = 1 : 2, PQ = 6 cm, PY = 4 cm, find PR and QR.

Answer 1

Given:

∠PQR = 90°, XY || QR.

PX : QX = 1 : 2

PQ = 6 cm

PY = 4 cm

Step-wise calculation:

1. From PX : QX = 1 : 2.

Write PX = k, QX = 2k.

So, PQ = PX + QX = 3k.

Thus, 3k = 6 ⇒ k = 2.

Therefore, PX = 2 cm and QX = 4 cm.

2. Because XY || QR, 

ΔPXY ∼ ΔPQR   ...(Corresponding angles equal)

3. Corresponding sides give `(PX)/(PQ) = (PY)/(PR)`. 

Substitute known values:

`2/6 = 4/(PR)` 

⇒ `1/3 = 4/(PR)` 

⇒ PR = 3 × 4

⇒ PR = 12 cm

4. In right ΔPQR right at Q use Pythagoras: 

PR² = PQ² + QR² 

So, QR² = PR² – PQ²

= 12² – 6²

= 144 – 36

= 108

Hence, `QR = sqrt(108)`

= `6sqrt(3) ≈ 10.392  cm`