Question

There are two circles each with radius 5 cm. The length of tangent AB is 26 cm. Find the length of tangent CD.

• A. 15 cm
• B. 21 cm
• C. 24 cm
• D. 16 cm

Answer 1

The correct option is C

24 cm

From the figure shown above,
AB = PO = 26 cm

Consider triangle CQP and DQO
∠ CQP = ∠ DQO (Vertically opposite angles)
∠ PCQ = ∠ODQ = 90∘ (Line joining the center to the tangent at the point of contact of the circle)
PC = OD (radius is equal)
therefore the two triangles are congruent using AAS postulate

$PQ=QO=\frac{26}{2}=13cm$

Similarly CQ = QD

Consider triangle PCQ,
Applying Pythagoras Theorem, we get,

${CQ}^2+{CP}^2={PQ}^2$

${CQ}^2+5^2={13}^2$

${CQ}^2$ = ${13}^2$ – $5^2=144$

CQ = 12 cm

Therefore, CD = 2CQ = 24 cm