A circle of radius 4 cm is drawn inscribed in a right angle triangle ABC, right angled at C. If AC = 12 cm, then the value of CB is:
16 cm
Given:
Radius of circle = 4 cm
AC = 12 cm;
∠ACB = 90°
Concept used:
The tangent makes a right angle with the radius of a circle at the point of contact.
If two tangents are drawn from an external point of the circle, then they are equal in length.
Formula used:
Pythagoras theorem:
H2 = P2 + B2
Where, P = perpendicular; B = base;
H = hypotenuse
(a + b)2 = a2 + b2 + 2ab
Calculation:
Quadrilateral PCRO is a square.
PC = CR = 4 cm
AR = (AC - CR) = (12 - 4) = 8 cm
AR = AQ = 8 cm
In △ABC
(AB)2 = (AC)2 + (BC)2
⇒ (8 + x)2 = 122 + (4 + x)2
⇒ 64 + x2 + 16x = 144 + 16 + x2 + 8x
⇒ 64 + 16x = 160 + 8x
⇒ 8x = (160 - 64) = 96
⇒ x = 96/8 = 12
BC = (4 + x) = 4 + 12 = 16 cm
∴ The correct answer is 16 cm.
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