Note: In almost all the problems below the implementation of the following theorem is important. **Theorem 3: The perpendicular from the center of a circle to a chord bisects the chord.**

Question 1: The radius of the circle is and the length of one of its chords is . Find the distance of the chord from the center.

Answer:

Refer to the adjoining diagram.

cm

Question 2: The radius of the circle is and the length of one of its chords is . Find the distance of the chord from the center.

Answer:

Refer to the adjoining diagram.

cm

Question 3: Find the length of a chord which is at a distance of from the center of the circle of radius .

Answer:

Refer to the adjoining diagram.

cm

Therefore cm

Question 4: Find the length of a chord which is at a distance of from the center of the circle of radius .

Answer:

Refer to the adjoining diagram.

cm

Therefore cm

Question 5: Find the length of a chord which is at a distance of from the center of the circle of radius .

Answer:

Refer to the adjoining diagram.

cm

Therefore cm

Question 6: Two chords and of length and respectively of a circle are parallel. If the distance between and is , find the radius of the circle.

Answer:

Refer to the adjoining diagram.

Squaring both sides we get

Squaring both sides we get

Question 7: An equilateral triangle of side is inscribed in a circle. Find the radius of the circle.

Answer:

Refer to the adjoining diagram.

cm

Question 8: is a diameter of the circle. is a point in such that and . Find the length of the shortest chord through .

Answer:

Refer to the adjoining diagram.

cm

cm

Therefore

Therefore cm

Hence cm

Question 9: The length of the common chord of two intersecting circles is . If the radii of the two circles are and , find the distance between their centers.

Answer:

Refer to the adjoining diagram.

Therefore

cm

Similarly,

Therefore

cm

Hence

cm

Question 10: A rectangle with a side of length is inscribed in a circle of diameter . Find the area of the rectangle.

Answer:

Refer to the adjoining diagram.

cm

cm

Hence the area of rectangle

Question 11: The center of a circle of radius units is the point . is a point inside the circle. is a chord of the circle such that . Calculate the length of .

Answer:

Refer to the adjoining diagram.

cm

Therefore cm

Hence cm

Question 12: and are two parallel chords of a circle whose center is and radius is . If is and is , find the distance between and , if they lie i) on the same side of center ii) on the opposite side of center

Answer:

Refer to the adjoining diagram.

i) cm

cm

Therefore cm

ii) cm

cm

Therefore cm

Question 13: and are two parallel chords of a circle such that and . If the chords are on the opposite sides of the center and the distance between then is , find the radius of the circle.

Answer:

Refer to the adjoining diagram.

Similarly,

Therefore

Squaring both sides

Squaring both sides

cm

Question 14: and are two chords of a circle such that and . . If the distance between and is , find the radius of the circle.

Answer:

Refer to the adjoining diagram.

Squaring both sides

Squaring both sides

cm

Question 15: is an isosceles triangle inscribed in a circle. If and , find the radius of the circle.

Answer:

Refer to the adjoining diagram.

cm

Squaring both sides

cm

Question 16: In a circle of radius , and are two chords such that . Find the length of the chord .

Answer:

Refer to the adjoining diagram.

Therefore

Squaring both sides

cm

Question 17: Two concentric circles with center have as the points of intersection with the line as shown in the diagram. If and , find the lengths of and .

Answer:

Refer to the adjoining diagram.

cm cm

cm cm

cm

cm

Question 18: Two circles of radii and intersect and the length of the common chord is . Find the distance between their centers.

Answer:

Refer to the adjoining diagram.

Similarly,

Therefore

cm

Question 19: In the figure, two circles with center and and of radii and touch each other internally. If the perpendicular bisector of segment meets the bigger circle in and , find the length of .

Answer:

Refer to the adjoining diagram.

Therefore cm