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 AP = x m.

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 circle.

Ans: (i) Length of AR:

By tangents property, we know that the tangents drawn on a circle from an external point are always equal

∴ AP = AR = X

Therefore, Length of AR is X m.

(ii) Type of quadrilateral BQOR:

Next let’s take quadrilateral BQOR. We have,

BQ = BR (being tangents drawn to same circle from an external point)

OR = OQ (being radii of same circle)

It is given that ∠ RBQ = 90⁰

Now, by tangents property, we know that the tangent make right angle to the radius of the circle at the point of contact.

Hence, ∠ QQB = 90⁰  and ∠ ORB = 90⁰

We know that sum of all angles of a quadrilateral will be equal to 360⁰. Therefore,

∠ RBQ + ∠ ORB + ∠ ROQ + ∠ OQB = 360⁰

90⁰ + 90⁰ + 90⁰ + ∠ ROQ = 360⁰

∠  ROQ = 90⁰

Since all the angles of the quadrilateral are equal to 90⁰ and the adjacent sides are also equal,

Therefore the quadrilateral BQOR is a square.

(iii) (a) Length of PC in terms of X

AP = AR = X 

BR = AB – AR = 7 – X

BQ = BR = 7 – X

QC = BC – BQ

= 15 – (7 – x) = 8 + X

CP = CQ = 8 + X

Therefore, length of PC in terms of x is (8 + X) m

(to find the value of X, pls refer solution of (iii) (b) (i) below).

(iii) (b) (i) Calculating value of x:

We can see in the diagram that AC = AP + CP

∴ AC = x + 8 + x

∴ AC = 8 + 2 x ………… (i)

In Δ ABC, by Pythagoras theorem, AB²  + BC²  = AC² 

AC =

=

=

=

= 16.553 cm …. (ii)

By comparing equations (i) and (ii), we get:

8 + 2 x = 16.553

∴ 2 X = 16.553 – 8 = 8.553

∴ X = 4.2765

Therefore, value of X is 4.2765 m

(iii) (b)(ii) Value of radius r of circle:

In Quadrilateral BQOR, RO = BQ = r (RO is the radius r of the circle,)

Earlier, we had calculate value of BQ = 7 – x

∴ r = 7 – x

= 7 – 4.2765

= 2.7235

Therefore, radius r of the circle is 2.7235 m

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