Question 21: Prove that: (i) the parallelogram inscribed in a circle is a rectangle (ii) the rhombus, inscribed in a circle is a square.

Answer:

(i) Let be a parallelogram inscribed in the circle.

(opposite angles of a parallelogram are equal)

Similarly,

Therefore is a rectangle

(ii) is a rhombus (given) i.e. all four sides are equal.

Similarly,

Therefore is a square

Question 22: In the following figure . Prove that is an isosceles trapezium.

Answer:

(given)

is a cyclic quadrialteral)

… … … … (i)

(corresponding angles)

Hence

… … … … (ii)

Hence because of (i) and (ii), is an isosceles trapezium.

Question 23: Two circles intersect at and . Through diameters and of the two circles are drawn. Show that the points and are collinear.

Answer:

(angle in a semi circle)

(angle in a semi circle)

Therefore

Therefore are collinear.

Question 24: is a quadrilateral inscribed in a circle. having . is the center of the circle. Show that: .

Answer:

(cyclic quadrilateral)

(sum of the angles in a triangle is 180)

Similarly

Therefore

Question 25: The figure given below shows the circle with center . Given  and . (i) Find the relationship between . (ii) Find  if is a parallelogram.

Answer:

(i) and (given)

(ii) If is a parallelogram

Therefore (opposite angles are equal)

(radius of the same circle)

Hence

Question 26: Two chords and intersect at inside the circle. Prove that the sum of the angles subtended by the arcs and at the center is equal to twice the .

Answer:

Similarly,

Adding the two

… … … … (i)

In  

… … … … (ii)

Using (1) and (ii)

Question 27: In the given figure is a diameter of the circle. and . Find (i)  (ii)

Answer:

(i) (angle in a semi circle)

(ii) Given

(alternate angles)

Question 28: In the given figure, and is the center of the circle. If , find the . Give reasons.

Answer:

and (angles in a semi circle)

Since

In

(cyclic quadrilateral)

(angles in the same segment)

Question 29: Two circles intersect at and . Through a straight line is drawn to meet the circles in and . Through , a straight line is drawn to meet the circles at and . Prove that is parallel to .

Answer:

and are cyclic quadrilateral

… … … … (i)

… … … … (ii)

  … … … … (iii)

From (i) and (ii)

  … … … … (iv)

From (iv) and (iii)

    … … … … (v)

Therefore by (v)

Question 30: is a cyclic quadrilateral in which and on being produced, meet at such that . Prove that is parallel to .

Answer:

is a cyclic quadrilateral, and (given)