Question : A circle is inscribed in a ΔABC having sides AB = 16 cm, BC = 20 cm, and AC = 24 cm, and sides AB, BC, and AC touch circle at D, E, and F, respectively. The measure of AD is:
Option 1: 10 cm
Option 2: 20 cm
Option 3: 6 cm
Option 4: 14 cm
Correct Answer: 10 cm
Solution :
Given:
A circle is inscribed in a ΔABC having sides AB = 16 cm, BC = 20 cm, and AC = 24 cm, and sides AB, BC, and AC touch circle at D, E, and F, respectively.
We know that tangents drawn from an external point to a circle are equal in length.
Let AD = AF = a, BD = BE = b and CE = CF = c
According to the question,
AB = a + b
⇒ 16 = a + b .......(1)
BC = b + c
⇒ 20 = b + c .......(2)
AC = a + c
⇒ 24 = a + c .......(3)
Now, On adding equations (1), (2), and (3), we get
⇒ 16 + 20 + 24 = a + b + b + c + a + c
⇒ 60 = 2(a + b + c)
⇒ a + b + c = 30
Now putting the value of (b + c) in the above equation,
⇒ a + 20 = 30
⇒ a = 10 cm
Hence, the correct answer is 10 cm.
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