Question 1: A chord of length is drawn in a circle of radius . Calculate its distance from the center of the circle.

Answer:

Let the distance from the center

Therefore

Question 2: A chord of length is drawn at a distance of from the center of a circle. Calculate the radius of the circle.

Answer:

Let the radius

Therefore

Question 3: The radius of a circle is and the length of perpendicular drawn from the center to a chord is . Calculate the length of the chord.

Answer:

Let the length of the chord

Therefore

Therefore the length of the chord

Question 4: A chord of length 24 cm is at a distance of 5 cm from the center of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the center.

Answer:

Let the radius

Therefore

Let the length of the chord

Therefore

Therefore the length of the chord

Question 5: In the following figure, is a straight line and is the center of both the circles. If and , find the length of .

Answer:

Therefore

Question 6: is the center of the circle of radius . is any point in the circle such that . is the point travelling along the circumference, is the distance from . What are the least and the greatest values of . What is the position of the points at these values . **[1992]**

Answer:

When is on , the least distance of

When is on extended , the greatest distance of

Question 7: In a circle of radius , two parallel chords of length and are drawn. Find the distance between the chords, if both the chord are:

(i) on the opposite side of the center

(ii) on the same side of the center

Answer:

(i) When the chords are on the opposite side of the triangles

Distance of the larger chord from the center

Distance of the smaller chord from the center

Therefore the distance between the chords

(ii) When the chords are on the same side of the triangles

Distance of the larger chord from the center

Distance of the smaller chord from the center

Therefore the distance between the chords

Question 8: Two parallel chords are drawn in a circle of diameter . The length of one chord is . and the distance between the two chords is . Find the length of another chord.

Answer:

Let the distance of the chord from the center

Therefore

Therefore the distance of the other chord from the center

Therefore the length of the other chord

Question 9: A chord of a circle whose center is , is bisected at by a diameter . Given and , calculate the lengths of (i) (ii) and (iii)

Answer:

Question 10: The figure given below, shows a circle with center in which Diameter bisects the chord at point . If and , find the radius of the circle.

Answer:

Let

Therefore

Therefore Radius

Question 11: The figure shows two concentric circles and is the chord of larger circle. Prove that .

Answer:

If you drop a perpendicular from to the chord, it would bisect the chord *( Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.)*. Let us say that it intersects and at .

Therefore and

If you subtract these two, we get

Question 12: A straight line is drawn cutting two equal circles and passing through the midpoint of the line joining their centers . Prove that the chords , which are intercepted by two circles are equal.

Answer:

First draw perpendiculars and

Now consider and

(opposite angles)

is the mid point of – Given)

Therefore

Therefore

Therefore *(Theorem 8 (Converse of Theorem 7) : Chords of circle which are equidistant from the center of the circle are equal in length.)*

Question 13: are mid points of two equal chords respectively of a circle with center . Prove that:

(i)

(ii)

Answer:

First drop perpendiculars and on and respectively.

will bisect and will bisect . *( Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.)*

Therefore

Given . Therefore

Also

Therefore in (since angles opposite equal sides of a triangle are equal)

We know

(i) Therefore

(ii) We know

Therefore

Question 14: In the following figure; and are the points of intersection of two circles with centers and . If straight lines and are parallel to ; prove that

(i)

(ii)

Answer:

… … … … (i)

… … … … (ii)

Therefore from (i) and (ii) we get . Hence proved.

Question 15: Two equal chords and of a circle with center , intersect each other at point inside the circle. Prove that:

(i)

(ii)

Answer:

Draw and

*(Theorem 5: A perpendicular to a chord, from the center of the circle, bisects the chord.)*

In

In

(radius of the circle)

Consider

is common

Therefore

(Given)

Since

Therefore

Question 16: In the following figure, is a square. A circle is drawn with as the center which meets at and at . Prove that:

(i)

(ii)

Answer:

(i) In

(radius)

(side of square)

Therefore (S.A.S postulate)

(ii) (radius)

(side of a square)

Therefore

Consider

(side of a square)

Therefore

Question 17: The length of common chord of two intersecting circles is . If the diameter of the circles be and , calculate the distance between their center.

Answer:

and (given)

In

Similarly in

Therefore

Question 18: The line joining the mid points of two chords if a circle passes through the center. Prove that the chords are parallel.

Answer:

(given)

bisects (given)

Therefore (if line drawn from the center bisects the chord, then is is perpendicular to the chord)

Similarly

Therefore (alternate angles)

Therefore

Question 19: In the following figure, the line ; where and are the centers of the circles. Show that:

(i)

(ii)

Answer:

… … … … (i)

… … … … (ii)

(i) – (ii) we get

. Hence proved.

Question 20: and are two equal chords of a circle with center which intersect each other at right angle at point . If and , show that is a square.

Answer:

and (given)

Since and and because all angles are we can say

Therefore is square.

Question 21: In the given figure, is the center of the circle. and are two chords of circle. and . , . Find the

(i) radius of the circle

(ii) length of chord ** [2014]**

Answer:

, ,

Therefore

(i) Consider

Radius of the circle.

(ii) Now consider