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Circles — Maths STD 10 — Question

Maharashtra BoardEnglish MediumSTD 10MathsCircles2 Marks
Question
In the given figure, a circle with centre O, is inscribed in a quadrilateral ABCD such that it touches the side BC, AB, AD and CD at points P, Q, R and S respectively. If AB = 29cm, AD = 23cm, $\angle\text{B} = 90^\circ$and DS = 5cm then find the radius of the circle.
✓

Answer

We know that tangent segments to a circle from the same external point are congruent.
Now, we have
DS = DR, AR = AQ
Now, AD = 23cm
⇒ AR + RD = 23
⇒ AR = 23 - RD
⇒ AR = 23 - 5 $[\therefore$ DS = DR = 5$]$
⇒ AR = 18cm
Again, AB = 29cm
⇒ AQ + QB = 29
⇒ QB = 29 - AQ
⇒ QB = 29 - 18 $[\therefore$ AR = AQ = 18$]$
⇒ QB = 11cm
Since all the angles are in a quadrilateral BQOP are right angles and OP = BQ.
Hence, BQOP is a square.
We know that all the sides of square are equal.
Therefore, BQ = PO = 11cm
Hence, the radius of the circle is 11cm.

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