Question 15: In the given figure, is the diameter of the circle with center . If , find .
Answer:
subtends at the center and at the circumference of the circle.
Question 16: In a Cyclic-quadrilateral . Sides and produced meet at point whereas sides and produced meet at point . lf ; find and .
Answer:
Given is a cyclic quadrilateral.
or if , then
(sum of the opposite angles in a cyclic quadrilateral is )
In
… … … (i)
In
Therefore … … … (ii)
Hence
Therefore
Question 17: In the following figure, is the diameter of a circle with center and is the chord with length equal to radius . If produced and produced meet at point ; show that .
Answer:
Given: is diameter,
In
is equilateral
Therefore
In
(radius of the same circle)
Therefore
Similarly in
(radius of the same circle)
Therefore
Since is cyclic quadrilateral
Therefore (opposite angles of a cyclic quadrilateral are supplementary)
In
. Hence proved.
Question 18: In the following figure, is a cyclic quadrilateral in which is parallel to . If the bisector of meets at point and the given circle at point , prove that: (i) (ii)
Answer:
Given: is a cyclic quadrilateral
is the angle bisector) … … … (i)
(i) (alternate angles) … … … (ii)
In
Using (i) and (ii) we get
(vertically opposite angles)
is a cyclic quadrilateral)
Also
(ii) subtends on circumference
subtends on circumference
Given
Therefore (equal arcs subtends equal angles on circumference)
Question 19: is a cyclic quadrilateral. Sides and produced meet at point ; whereas sides and produced meet at point . If , find the angles of the cyclic quadrilateral .
Answer:
Given
In
In
or
Therefore
Question 20: In the following figure shows a circle with as its diameter. If and , find the perimeter of the cyclic quadrilateral . [1992]
Answer:
Given:
Therefore the perimeter of
Question 21: In the following figure, is the diameter of a circle with center . If , prove that: (i) (ii) is bisector of . Further, if the length of , find : (a) (b) .
Answer:
Given:
Consider and
is common
(angles in semi circle)
Therefore
(i) (corresponding parts of congruent triangles)
(ii) (equal chords subtend equal angles on the circumference of the same circle)
Therefore bisects
Question 22: In cyclic quadrilateral ; and ; find: (i) (ii) (iii) (iv)
Answer:
Given and
(angles in the same segment)
(angles in the same segment)
(opposite angles in a cyclic quadrilateral are supplementary)
Therefore
Question 23: In the given figure and ; find the values of and . [2007]
Answer:
Given and
(ABDE is a cyclic quadrilateral)
(straight line)
In :
Therefore
Therefore is a cyclic quadrilateral)
In :
Question 24: In the given figure, and . Find (i) (ii) (iii)
Answer:
Given and . Also
Therefore (alternate angles)
Hence
Therefore
and
Therefore
In :
Therefore
(angles in the same segment)
Question 25: is a cyclic quadrilateral of a circle center such that is a diameter of the circle and the length of the chord is equal to the radius of the circle. If and produced meet at , show that .
Answer:
Given
In :
In (Radius of the same circle)
Let
Therefore … … … (i)
In (Radius of the same circle)
Let
Therefore … … … (ii)
Now, (straight line angle)
In :
Question 26: In the figure, given alongside,
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