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 .

Two Parallel Tangents to a Circle

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.

Tangents from an External Point

Answer

Join OE and OF. Also join OA, OB, and OC.

Tangents from an External Point

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