Question 1: A chord of length $6 \ cm$ is drawn in a circle of radius $5 \ cm$. Calculate its distance from the center of the circle.

Let the distance from the center $= x$

Therefore $x = \sqrt{5^2-3^2}= \sqrt{25-9} = 4 \ cm$

$\\$

Question 2: A chord of length $8 \ cm$ is drawn  at a distance of $3 \ cm$ from the center of a circle. Calculate the radius of the circle.

Let the radius $= r \ cm$

Therefore $r = \sqrt{4^2+3^2} = 5 \ cm$

$\\$

Question 3: The radius of a circle is $17 \ cm$ and the length of perpendicular drawn from the center to a chord is $8 \ cm$. Calculate the length of the chord.

Let the length of the chord $= 2x$

Therefore $x = \sqrt{17^2-8^2} =\sqrt{289-64} = 15 \ cm$

Therefore the length of the chord $= 2x = 30 \ cm$

$\\$

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.

Let the radius $= r \ cm$

Therefore $r = \sqrt{12^2+5^2} = 13 \ cm$

Let the length of the chord $= 2x$

Therefore $x = \sqrt{13^2-12^2} =\sqrt{169-144} = 5 \ cm$

Therefore the length of the chord $= 2x = 10 \ cm$

$\\$

Question 5: In the following figure, $AD$ is a straight line $OP \perp AD$ and $O$ is the center of both the circles. If $OA = 34 \ cm, OB = 20 \ cm$ and $OP = 16 \ cm$, find the length of $AB$.

$BP = \sqrt{20^2-16^2}=\sqrt{400-256}= 12 \ cm$

$AP = \sqrt{34^2-16^2}=\sqrt{1156-256}= 30 \ cm$

Therefore $AB = (AP-BP) = 30 - 12 = 18 \ cm$

$\\$

Question 6: $O$ is the center of the circle of radius $10 \ cm$. $P$ is any point in the circle such that $OP=6 \ cm$. $A$ is the point travelling along the circumference, $x$ is the distance from $A to P$. What are the least and the greatest values of $x \ in \ cm$. What is the position of the points $O, P \ and \ A$ at these values . [1992]

When $P$ is on $OA$, the least distance of $x = 4 \ cm$

When $P$ is on extended $OA$, the greatest distance of $x = 16 \ cm$

$\\$

Question 7: In a circle of radius $17 \ cm$, two parallel chords of length $30 \ cm$ and $16 \ cm$ 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

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

Distance of the larger chord from the center $x_1 = \sqrt{17^2-15^2}= 8 \ cm$

Distance of the smaller chord from the center $x_2 = \sqrt{17^2-8^2}= 15 \ cm$

Therefore the distance between the chords $= x_1 + x_2 = 23 \ cm$

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

Distance of the larger chord from the center $x_1 = \sqrt{17^2-15^2}= 8 \ cm$

Distance of the smaller chord from the center $x_2 = \sqrt{17^2-8^2}= 15 \ cm$

Therefore the distance between the chords $= x_2 - x_1 = 7 \ cm$

$\\$

Question 8: Two parallel chords are drawn in a circle of diameter $30 \ cm$. The length of one chord is $24 \ cm$. and the distance between the two chords is $21 \ cm$. Find the length of another chord.

Let the distance of the $24 \ cm$ chord from the center $= x \ cm$

Therefore $x = \sqrt{15^2-12^2}= 9 \ cm$

Therefore the distance of the other chord from the center $= 21 - 9 = 12 \ cm$

Therefore the length of the other chord $= 2 \times \sqrt{15^2-12^2} = 18 \ cm$

$\\$

Question 9: A chord $CD$ of a circle whose center is $O$, is bisected at $P$ by a diameter $AB$. Given  $OA = OB = 15 \ cm$ and $OP = 9 \ cm$, calculate the lengths of  (i) $CD$ (ii) $AD$ and (iii) $CB$

$CP = \sqrt{15^2-9^2} = 12 \ cm$

$CD = 2 \times 12 = 24 \ cm$

$AD = \sqrt{24^2+12^2}=\sqrt{720} = 26.83 \ cm$

$CB = \sqrt{12^2+(15-9)^2} = \sqrt{144+36} = 13.42 \ cm$

$\\$

Question 10: The figure given below, shows a circle with center $O$ in which Diameter $AB$ bisects the chord $CD$ at point $E$. If $CE=ED=8 \ cm$ and $EB=4 \ cm$, find the radius of the circle.

Let $OE = x \ cm$

Therefore

$x^2+8^2 = (x+4)^2$

$x^2 + 64 = x^2 + 8x + 16$

$8x = 48 \Rightarrow x = 6 \ cm$

Therefore Radius $= 6 + 4 = 10 \ cm$

$\\$