Question

In the given figure, a circle inscribed in \( \triangle ABC \) touches \( AB, BC, \) and \( CA \) at \( X, Z, \) and \( Y \) respectively.

If \( AB = 12 \, \text{cm}, AY = 8 \, \text{cm}, \) and \( CY = 6 \, \text{cm} \), then the length of \( BC \) is:
 

• A. 14 cm
• B. 12 cm
• C. 10 cm
• D. 8 cm

Hint:

When a circle is inscribed in a triangle, the lengths from any vertex to the points of tangency with adjacent sides are equal. Use tangent-length equality to find unknown sides.

Answer 1

The Correct Option is C

Step 1: Let the points where the incircle touches the sides be:
- \( X \) on \( AB \)
- \( Y \) on \( AC \)
- \( Z \) on \( BC \)
Step 2: Use the property of tangents drawn from an external point: Tangents from the same external point to a circle are equal in length.
Step 3: Assign tangent lengths based on given data:
- From point \( A \): \[ AX = AY = 8 \, \text{cm} \]
- From point \( C \): \[ CZ = CY = 6 \, \text{cm} \]
- From point \( B \): \[ BX = BZ \]
Step 4: Use the given length \( AB = 12 \, \text{cm} \): \[ AB = AX + BX = 8 + BX \Rightarrow BX = 12 - 8 = 4 \, \text{cm} \]
So: \[ BZ = 4 \, \text{cm} \]
Step 5: Find the length of \( BC \): \[ BC = BZ + ZC = 4 + 6 = 10 \, \text{cm} \]
Final Answer:
\[ \boxed{BC = 10 \, \text{cm}} \]