Chord Properties of a Circle — Competency Focused Questions

Two chords of a circle of lengths 10 cm and 8 cm are at the distances of 6 cm and 5 cm respectively from the centre. This statement is :

1. True

2. False

3. Can't say anything

4. Data inadequate

Answer

The statement is false because there is an inverse relationship between the chord length and its distance from the center :

The longer the chord, the closer it is to the center.

The shorter the chord, the farther it is from the center.

Hence, option 2 is the correct option.

The radius of a circle is 13 cm and length of chord is 10 cm. The shortest distance between chord and the centre is:

1. 12 cm

2. 15 cm

3. 16 cm

4. 18 cm

Answer

Let AB be chord of length = 10 cm

Radius OA = 13 cm

Let OD be the perpendicular distance from the center to the chord, it bisects the chord into two equal parts:

AD = DB = $\dfrac{10}{2}$ = 5 cm

In triangle OAD,

According to the Pythagorean Theorem:

OA² = OD² + AD²

13² = OD² + 5²

169 = OD² + 25

OD² = 169 - 25

OD² = 144

OD = $\sqrt{144}$ = 12 cm

The shortest distance between the chord and the center is 12 cm.

Hence, option 1 is the correct option.

In the circle, chord AB of length 12 cm is bisected by diameter CD at P, so that CP = 3 cm. Radius of the circle is:

1. 5.2 cm

2. 6.5 cm

3. 7.5 cm

4. 8.3 cm

Answer

Given,

Chord AB = 12 cm

CD bisects chord AB at P

AP = PB = $\dfrac{12}{2}$ = 6 cm

CP = 3 cm

OC = OA = R

OP = OC - CP = R - 3

In triangle OPA,

Apply the Pythagorean Theorem :

OA² = OP² + AP²

R² = (R - 3)² + 6²

R² = R² - 6R + 9 + 36

R² = R² - 6R + 45

6R = 45

R = $\dfrac{45}{6}$ = 7.5 cm

Hence, option 3 is the correct option.

Three friends Amit, Vinay and Sukrit are playing a game by standing on a circle of radius 5 m drawn in a park. Amit throws a ball to Vinay, Vinay to Sukrit, Sukrit to Amit. If the distance between Amit and Vinay and between Vinay and Sukrit is 6 m each, what is the distance between Amit and Sukrit?

Answer

Let the center of the circular park be O and positions of Amit, Vinay and Sukrit be A, V and S respectively.

The radius of the circle is 5 m.

AV = VS = 6 m

Draw VM ⊥ AS.

In an isosceles triangle, the perpendicular from a vertex between equal sides bisects the opposite side.

∴ AM = MS

Thus,

Let OM = x m and MS = y m

VM = OV - OM = (5 - x) m.

In right-angled triangle VMS,

⇒ VS² = VM² + MS²

⇒ 6² = (5 - x)² + y²

⇒ 36 = (5 - x)² + y²

⇒ y² = 36 - (5 - x)² ....(1)

In right-angled triangle OMS,

⇒ OS² = OM² + MS²

⇒ 5² = x² + y²

⇒ y² = 25 - x²....(2)

From (1) and (2), we get :

⇒ 25 - x² = 36 - (5 - x)²

⇒ 25 - x² = 36 - (25 - 10x + x²)

⇒ 25 - x² = 36 - 25 + 10x - x²

⇒ 25 - x² = 36 - 25 + 10x - x²

⇒ 25 = 11 + 10x

⇒ 10x = 14

⇒ x = 1.4

Substituting value of x in equation (2):

⇒ y² = 25 - (1.4)²

⇒ y² = 23.04

⇒ y = $\sqrt{23.04}$ = 4.8 m

From figure,

⇒ AS = MS + AM

⇒ AS = 2MS

⇒ AS = 2(4.8) = 9.6 m

Hence, the distance between Amit and Sukrit is 9.6 m.

Show that the diameter is the greatest chord of a circle.

Answer

Let AB be a diameter of the circle and O be the centre.

By definition, it passes through the center, so AB = OA + OB = r + r = 2r.

Let CD be any other chord of the circle that does not pass through the center.

∴ OC = OD = r

The sum of the lengths of any two sides must be greater than the length of the third side.

OC + OD > CD

2r > CD

AB > CD

This proves the length of the diameter is always greater than the length of any other chord that does not pass through the center.

Hence, the diameter is the greatest chord of a circle.