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

Question

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 $AP = x$ m.

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Based on the above information, answer the following questions:
(i) Find the length of $AR$ in terms of $x$.
(ii) Write the type of quadrilateral $BQOR$.
(iii) (a) Find the length $PC$ in terms of $x$ and hence find the value of $x$.

OR

(b) Find $x$ and hence find the radius $r$ of the circle.

Accepted Answer

Step 1: (i) $AR = AP = x$ m (tangents from external point A)
Step 2: (ii) Quad. $ORBQ$ is a square.
Step 3: (iii) (a) $PC = 8 + 2x$
$AC^2 = (8 + 2x)^2 = 49 + 225 = 274$
$\Rightarrow 8 + 2x = \sqrt{274}$
Step 4: $\Rightarrow x = \frac{-8 + \sqrt{274}}{2}$ or $4.28$ approx.

Question

Solution