Two circles having radii 12 cm and 8 cm, respectively, touch each other externally. A common tangent is drawn to these circles which touch the circles at M and N, respectively. What is the length (in cm) of MN?
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. 8√6.
The common tangent MN is equal to PS
Triangle OSP is a right-angled triangle
PSMN is a rectangle. (All four interior angles of quadrilateral PSMN are right angles. Since tangent is perpendicular to the radius at the point of contact; and MS is also perpendicular to ON)
⇒ OS = OM - SM = 12 - 8 = 4cm
From the Pythagorean Theorem -
⇒ OP2 = OS2 + SP2
⇒ SP2 = OP2 - OS2
⇒ SP2 = 202 - 42 [OS = 4 cm and OP = 12 + 8 = 20cm]
⇒ SP2 = 384
⇒ SP = 8√6 cm
Also, the opposite sides of a rectangle are equal
Therefore the length of common tangent MN = 8√6 cm