Question

Two circles of radii 25cm and 9cm touch each other externally. Find the length of the direct common tangent. [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 3 Marks

Let the two circles with centers A and B and radii 25 cm and 9 cm respectively touch each other externally at a point C.

Then, AB = AC + CB = (25 + 9) cm = 34 cm.

Let PQ be a direct common tangent to the two circles.

Join AP and BQ.

Then, $AP\perp PQ$ and $BQ\perp PQ$.

[Radius through point of contact is perpendicular to the tangent]

Draw, $BL\perp AP$.

Then, PLBQ is a rectangle.

Now, LP = BQ = 9 cm and PQ = BL.

AL = (AP - LP) = (25 – 9) cm = 16 cm.

From right triangle ALB, we have

${AB}^2={AL}^2+{BL}^2\Rightarrow {BL}^2={AB}^2-{AL}^2={\left(34\right)}^2-{\left(16\right)}^2=(34+16)(34–16)=900$

$\Rightarrow BL=\sqrt{900}=30cm$.

PQ = BL = 30 cm.

Hence, the length of direct common tangent is 30 cm.