CBSE Class 10-Mathematics: Questions and Answers Chapter – 10 Circles Part 19 (For CBSE, ICSE, IAS, NET, NRA 2022)
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4 Mark Questions
Question 1:
In figure, XY and X ‘Y’ are two parallel tangents to a circle with center O and another tangent AB with point of contact C intersecting XY at A and X ‘Y’ at B. Prove that .
Answer:
Given: In figure, XY and X ‘Y’ are two parallel tangents to a circle with center O and another tangent AB with point of contact C intersecting XY at A and X ‘Y’ at B.
To Prove:
Construction: Join
Proof:
[Tangent at any point of a circle is to the radius through the point of contact]
In right angled triangles OPA and OCA,
[Common]
[Tangents from an external point to a circle are equal]
[RHS congruence criterion]
[By C. P. C. T.]
… (iii)
Similarly,
… (iv)
and a transversal AB intersects them.
[Sum of the consecutive interior angles on the same side of the transversal is ]
… (v)
[From eq. (iii) & (iv) ]
In ,
[Angel sum property of a triangle]
[From eq. (v) ]
Question 2:
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact are of lengths 8 cm and 6 cm respectively (see figure) . Find the sides AB and AC.
Answer
Join OE and OF. Also join OA, OB, and OC.
Since
[Tangents from an external point to a circle are equal]
Since
[Tangents from an external point to a circle are equal]
Let
Since
[Radii of a circle are equal]
Semi-perimeter of
Now, Area of of of
Squaring both sides,
And