A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that Find the length PC in terms of and hence find the

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Solution

Q.
A backyard is in the shape of a triangle ABC with right angle at B. AB = 7 m and BC 15 m. A circular pit was dug inside it such that it touches the walls AC, BC and AB at P, Q and R respectively such that $A P = x m$ .

Based on the above information, answer the question:

Q. Find the length PC in terms of $x$ and hence find the value of $x$

Ans.
(4.28)

Here, BC = 15 m

BQ = (7 – x)m

$∴$ QC = 15 – (7 – x)

or, QC = (8 + x)m

Also, QC = PC (Tangents from an external points C to the circle)

i.e., PC = (8 + x)m

In right DABC, using Pythagoras theorem,

$A C^{2} = A B^{2} + B C^{2}$

$\Rightarrow A C^{2} = 7^{2} + 15^{2} = 49 + 225 = 274$

$\Rightarrow A C = 16.55$

$\Rightarrow A P + P C = 16.55$

$\Rightarrow x + 8 + x = 16.55$

$\Rightarrow 2 x = 8.55$

$\Rightarrow x = 4.275 ~ 4.28 m$

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