Question : A circle of radius 4 cm is drawn in a right-angle triangle ABC, right-angled at C. If AC = 12 cm, then the value of CB is:
Option 1: 8 cm
Option 2: 12 cm
Option 3: 20 cm
Option 4: 16 cm
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Correct Answer: 16 cm
Solution :
OX = OY = 4 cm
CY = OX = 4 cm and CX = OY = 4 cm
CX = CY = 4 cm
Now, AC = 12 cm
AX = 12 – 4 = 8 cm
We know that tangents from external point to the circle are equal in length.
$\therefore$ AX = AZ = 8 cm
Let BY = $x$ cm
⇒ BZ = $x$ cm
Applying Pythagoras theorem,
$AB^2 = AC^2 + BC^2$
⇒ $(AZ+BZ)^2 = 12^2 + (BY+CY)^2$
⇒ $(8+x)^2=144+(4+x)^2$
⇒ $64+x^2 + 16x = 144 + 16 + x^2 + 8x$
⇒ $64+16 x = 160 + 8x$
⇒ $160-64 = 16x-8x$
⇒ $8x=96$
⇒ $x=12$ cm
$\therefore$ Length of CB = 12 + 4 = 16 cm
Hence, the correct answer is 16 cm.
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