Question

Three circles with centres O, P and Q touch each other externally. If their radii are 5 cm, 6 cm & 7 cm respectively, find the perimeter of \(\triangle \text{OPQ}\) :

• A. 36 cm
• B. 40 cm
• C. 35 cm
• D. 42 cm

Hint:

When circles touch externally, the distance between their centers is the sum of their radii. Let radii be \(r_O=5, r_P=6, r_Q=7\). The sides of \(\triangle \text{OPQ}\) are:
OP = \(r_O + r_P = 5 + 6 = 11\) cm
PQ = \(r_P + r_Q = 6 + 7 = 13\) cm
QO = \(r_Q + r_O = 7 + 5 = 12\) cm Perimeter = Sum of sides = \(11 + 13 + 12 = 36\) cm.

Answer 1

The Correct Option is A

Concept: When two circles touch each other externally, the distance between their centres is equal to the sum of their radii. The perimeter of a triangle is the sum of the lengths of its three sides. Step 1: Define the radii of the three circles Let the radii of the circles with centres O, P, and Q be \(r_O, r_P,\) and \(r_Q\) respectively. Given:
\(r_O = 5 \text{ cm}\)
\(r_P = 6 \text{ cm}\)
\(r_Q = 7 \text{ cm}\) Step 2: Determine the lengths of the sides of \(\triangle \text{OPQ}\) The vertices of the triangle are the centres of the circles O, P, and Q. Since the circles touch each other externally:
Side OP: This is the distance between the centres of the circles O and P. Since they touch externally, OP = \(r_O + r_P\). \[ OP = 5 \text{ cm} + 6 \text{ cm} = 11 \text{ cm} \]
Side PQ: This is the distance between the centres of the circles P and Q. Since they touch externally, PQ = \(r_P + r_Q\). \[ PQ = 6 \text{ cm} + 7 \text{ cm} = 13 \text{ cm} \]
Side QO (or OQ): This is the distance between the centres of the circles Q and O. Since they touch externally, QO = \(r_Q + r_O\). \[ QO = 7 \text{ cm} + 5 \text{ cm} = 12 \text{ cm} \] So, the lengths of the sides of \(\triangle \text{OPQ}\) are 11 cm, 13 cm, and 12 cm. Step 3: Calculate the perimeter of \(\triangle \text{OPQ}\) The perimeter of a triangle is the sum of its side lengths. Perimeter of \(\triangle \text{OPQ}\) = OP + PQ + QO Perimeter = \(11 \text{ cm} + 13 \text{ cm} + 12 \text{ cm}\) Perimeter = \(24 \text{ cm} + 12 \text{ cm}\) Perimeter = \(36 \text{ cm}\) The perimeter of \(\triangle \text{OPQ}\) is 36 cm. This matches option (1).