Question

2. Length of a chord of a circle is 32 cm, radius of the circle is 20 cm, find the distance
of the chord from the center of the circle.
​

Answer 1

Step-by-step explanation:

Handwriting is bad , but it should help

Answer 2

Given :

• Length of a chord (AB)  = 32 cm
• Radius of the circle (AO)  = 20 cm

To find :

• Distance of the chord from the centre of the circle

Concept :

• Perpendicular from the centre of the circle bisects the chord.

$\leadsto \bf{Pythagoras  \:  \: Theorem,} \\  \\  \sf \quad \rightarrow H^{2} =   P^{2} +  B^{2}$

where,

• H = Hypotenuse of the triangle which is also the longest side and is opposite to the right angle.
• P = Perpendicular of the triangle
• B = Base of the triangle

Solution :

In ∆AOP

→ 2AP = AB [∵ Perpendicular from the centre bisects the chord.]

→ 2AP = 32

→ AP = 32 ÷ 2

→ AP = 16

• Length of AP = 16 cm

In ∆AOP

By pythagoras theorem,

$\dashrightarrow \quad \sf H^{2} =   P^{2} +  B^{2}$

We have,

• H = AO = 20 cm
• P = OP = ?
• B = AP = 16 cm

$\\ \dashrightarrow \quad \sf (20)^{2} =   OP^{2} +  (16)^{2}$

Transposing 16² to the other side. [It's sign will get changed from (+) to (-).]

$\\ \dashrightarrow \quad \sf (20)^{2} - (16)^{2}=   OP^{2}$

Using Identity :-

$\\ \dashrightarrow \underline{\quad \sf \blue{(x - y)(x + y) = {x}^{2} - {y}^{2}}}$

$\\ \dashrightarrow \quad \sf (20 - 16)(20 + 16)=   OP^{2}$

$\\ \dashrightarrow \quad \sf (4)(36)=   OP^{2}$

Taking square root on both the sides.

$\\ \dashrightarrow \quad \sf \sqrt {(4)(36)}=   OP$

$\\ \dashrightarrow \quad \sf \sqrt {2 \times 2 \times 6 \times 6}=   OP$

$\\ \dashrightarrow \quad \sf 2 \times 6 =   OP$

$\\ \dashrightarrow \quad \sf 12 =   OP$

$\\ \dashrightarrow \pmb{ \quad \sf \purple{ Length \: of \:OP = 12  }}$

• Distance of the chord from the centre of the circle = 12 cm

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Know More →

About pythagoras theorem →

Pythagoras theorem :-

• In a right - angled triangle, the sum of the square of the hypotenuse is equal to the sum of the square of the other two sides.

Converse of pythagoras theorem :-

• If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right angled triangle.

About circle :- [Theorems]

• The angle at the centre is twice the angle at the circumference.
• Equal chords subtend equal angles at the centre.
• If the two angles subtended by the chords at the centre of the circle are equal, then the chords are also equal.
• The straight line passing through the centre of the circle bisects the chord and is perpendicular to it.