Question

In the following figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is ______.

Options

• 6 cm

• 9 cm

• 4 cm

• 2 cm

Answer 1

In the following figure, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is 4 cm.

Explanation:

Given that, PR = 12 cm, QR = 6 cm and PL = 8 cm

Now, In right angled ΔPLR, using Pythagoras theorem,

(Hypotenuse)² = (Perpendicular)² + (Base)²

⇒ PR² = PL² + LR²

⇒ LR² = PR² – PL² = (12)² – (8)²

⇒ LR² = 144 – 64 = 80

⇒ LR = `sqrt(80) = 4sqrt(5)` cm

∵ LR = LQ + QR

⇒ LQ = LR – QR = `(4sqrt(5) - 6)` cm

Now, area of ΔPLR,

`A_1 = 1/2 xx LR xx PL`

= `1/2 xx (4sqrt(5)) xx 8`

= `16sqrt(5)` cm²

Again, area of ΔPLQ,

`A_2 = 1/2 xx LQ xx PL`

= `1/2 xx (4sqrt(5) - 6) xx 8`

= `(16sqrt(5) - 24)` cm²

∴ Area of ΔPLR = Area of ΔPLQ + Area of ΔPQR

⇒ `16sqrt(5) = (16sqrt(5) - 24)` + Area of ΔPQR

⇒ Area of ΔPQR = 24 cm²

⇒ `1/2 xx PR xx QM = 24`

⇒ `1/2 xx 12 xx QM = 24`

∴  QM = 4 cm