Q) In figure, a circle is inscribed in a Δ ABC having sides BC = 8 cm, AB = 10 cm and AC = 12 cm. Find the length BL, CM, and AN.

Ans:

Step 1: Let’s consider BL = X

CL = BC – BL

∴ CL = 8 – X ………(i)

Step 2: Since, BL and BN are the tangents on the circle from same point B, these are equal.

∴ BN = BL = X

Step 2:

∵ AN = AB – BN

∴ AN =10 – X

Step 4: Since, AM and AN are the tangents on the circle from same point A, these are equal.

∴ AM = AN

∴ AM = 10 – X

Step 5: CM = AC – AM

∴ CM = 12 – (10 – X)

∴ CM = 2 + X

Step 5: Since, CM and CL are the tangents on the circle from same point C, these are equal.

∴ CL = CM

∴ CL = 2 + X ……. (ii)

Step 6: Next, we compare values of CL from equations (i) and (ii), we get:

8 – X = 2 + X

∴ 2 X = 6

∴ X = $\frac{6}{2} = 3

Therefore, the length of BP is 3 cm

Step 7: Since, Value of CM = 2 + X   (calculated in step 5)

∴ CM = 2 + 3 = 5

Therefore, the length of CM is 5 cm

Step 8: Since, Value of AN = 10 – X   (calculated in step 2)

∴ AN = 10 – 3 = 7

Therefore, the length of AN is 7 cm

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