Question

AB and CD are two parallel chords oflengths 8 cm and 6 cm
respectively. If they are 1 cm apart and lie on the same side of the centre,
find the distance of CD from the centre.



Please ans fast​

Answer 1

Answer:

Given,

AB=8cm

CD=6cm

PQ=1cm

Let radius of the circle be r

Therefore,

OA=OC=r

We know that the perpendicular dripped from the centre of the circle on the chord bisects the chord.

Therefore,

CQ=QD=3cm

AP=PB=4cm

Let OP=x

=>OQ=x+1

Now,

OA

2

=OP

2

+AP

2

=>r

2

=x

2

+4

2

=>r

2

=x

2

+16 (i)

OC

2

+OQ

2

+CQ

2

=>r

2

=(x+1)

2

+3

2

=>r

2

=x

2

+2x+1+9

=>x

2

+16=x

2

+2x+1+9 (from (i)

=>2x=6

=>x=3

Therefore,

r

2

=3

2

+16

=>r

2

=9+16

=>r

2

=25

=>r=5cm

Answer 2

What can we see inside?

We observe that the two triangles have the radius.

Radii are $\sf{\overline{OA}}$ and $\sf{\overline{OC}}$.

The distance of $\sf{\overline{CD}}$ is $\sf{\overline{ON}}$.

Solving for ΔOCN

Let's solve an equation to find the height $\sf{\overline{ON}}$.

We have $\sf{\overline{ON}=x}$ cm

Now any line perpendicular to the chord will bisect it.

We have $\sf{\overline{OA}=3}$ cm

• $\sf{\overline{ON}^2+\overline{CN}^2=\overline{OC}^2}$ [Pythagorean Theorem]

$\sf{\overline{OC}^2=x^2+3^2}$ ...(B)

Solving for ΔOAM

We have $\sf{\overline{OM}=1-x}$ cm

Any line perpendicular to the chord will bisect it.

We have $\sf{\overline{AM}=4$ cm

• $\sf{\overline{AM}^2+\overline{OM}^2=\overline{OA}^2}$ [Pythagorean Theorem]

$\sf{\overline{OA}^2=(x-1)^2+4^2}$

$\sf{\overline{OA}^2=x^2-2x+17}$ ...(C)

Two radii are equal.

According to (A) and (B)

$\sf{\cancel{x^2}+3^2=\cancel{x^2}-2x+17}$

$\sf{2x=8}$

∴$\sf{x=4}$ cm

Therefore the height is $\sf{\overline{ON}=4}$ cm.

Advice

• Steps for geometry

1. Use the diagram.
2. Use the given information to construct equations.
3. After we solve equations, we get the answer.

• Commonly used Circle Properties

1. A tangent to a circle is at a right angle.
2. The radius perpendicular to a chord bisects the chord.
3. Two tangents from a point outside are equal in length.