Question

In Fig, if PR = 12 cm, QR = 6 cm and PL = 8 cm, then QM is

(a) 6 cm (b) 9 cm (c) 4 cm (d) 2 cm

Answer 1

(c) 4 cm

From the question it is given that, PR = 12 cm, QR = 6 cm, PL = 8 cm

We know that, Pythagoras theorem

Hypotenuse² = perpendicular² + base²

In the given figure,

PR² = PL² + LR²

LR² = PR² – PL²

LR² = 12² – 8²

LR² = 144 – 64

LR² = 80

By transferring the square it becomes,

LR = √80

Therefore, LR = 4√5

Then,

LR = LQ + QR

LQ = LR – QR

LQ = 4√5 – 6

We know that, area of triangle PLR = ½ × LR × PL

= ½ × 4√5 × 8

= 1× 4√5 × 4

= 16√5

Now, area of triangle PLQ = ½ × LQ × PL

= ½ × (4√5 – 6) × 8

= 4(4√5 – 6)

= 16√5 – 24 cm²

So, Area of PLR = Area of PLQ + Area of PQR

16√5 = (16√5 – 24) + Area of PQR

Therefore, Area of PQR = 24 cm²

½ × PR × QM = 24

½ × 12 × QM = 24

6 × QM = 24

QM = 24/6

QM = 4 cm