CBSE+NCERT+10th- Math-Chapter-10-Circles-test-paper

1. Prove that the parallelogram circumscribing a circle is rhombus.

Ans Given : ABCD is a parallelogram circumscribing a circle.

To prove : - ABCD is a rhombus

or

AB=BC=CD=DA

Proof: Since the length of tangents from external are equal in length

AS = AR …..(1)

BQ = BR …..(2)

QC = PC …..(3)

SD = DP …..(4)

Adding (1), (2), (3) & (4).

AS + SD + BQ + QC = AR + BR + PC + DP

AD + BC = AB + DC

AD + AD = AB + AB

Since BC = AD & DC = AB (opposite sides of a parallelogram are equal)

2AD = 2AB

AD = AB …..(5)

BC = AD (opposite sides of a parallelogram)

DC = AB …..(6)

From (5) and (6)

AB = BC = CD = DA

2. A circle touches the side BC of a triangle ABC at P and touches AB and AC when

produced at Q and R respectively as shown in figure. Show that AQ= 1/2(perimeter of triangle ABC)

3. XP and XQ are tangents from X to the circle with centre O. R is a point on the circle. Prove that XA+AR=XB+BR

4.  In figure, the incircle of triangle ABC touches the sides BC, CA, and AB at D, E, and F respectively. Show that AF+BD+CE=AE+BF+CD= 1/2(perimeter of triangle ABC),

5. 5. A circle touches the sides of a quadrilateral ABCD at P, Q, R and S respectively. Show that the angles subtended at the centre by a pair of opposite sides are supplementary.

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