Two circles with centers B and O have radii OA = 8 cm and BC = x cm, respectively. AC is tangent to both circles. If OB and AC intersect the point E, AE = 12 cm and EC = 18 cm, then find the value of x (in cm).
Show answer & explanation
Correct answer: A
Explanation
The correct answer is Option 1 i.e. 12.
The radius of a small circle (r) = 8 cm
AE = 12 cm
EC = 18 cm
The tangent makes a right angle with the radius of a circle at the point of contact.
The vertically opposite angles are equal.
AO ⊥ AC and BC ⊥ AC
⇒ ∠ OAE = ∠ BCE = 90°
In Δ OAE and Δ BCE
⇒ ∠ OAE = ∠ BCE = 90°
⇒ ∠ AED = ∠ BEC (Vertically opposite angle)
⇒ Δ OAE ~ Δ BCE (By AA similarity)
By Corresponding Parts of Congruent Triangles -
⇒ AE/EC = AO/BC
⇒ 12/18 = 8/BC
⇒ BC = 144/12 = 12 cm