Question

AB and CD are two parallel chords of a circle which are on the opposite sides of the centre such that ab=16 cm and cd=12 cm and the distance between them is 14 cm. Find the radius of the circle

Answer 1

Pythagorean theorem,
8^2+x^2=r^2
64+36=r^2
r=10

Answer 2

The radius of the circle is 10 cm.

Step-by-step explanation:

Given,

AB and CD are two parallel chords of a circle with center O.

Length of AB = 16 cm

Length of CD = 12 cm

We have to find out the radius of the circle.

Solution,

We have drawn the circle for your reference.

And also given length of MN = 14.

Let the radius of the circle be 'r'.

And let the length of ON be 'x'.

AN = $\frac{AB}{2}=\frac{16}{2} =8\ cm$

Again, CM = $\frac{CD}{2}=\frac{12}{2} =6\ cm$

Now, In ΔANO

By Pythagoras theorem, square of the hypotenuse is equal to the sum of squares of the other two sides of triangle.

$AN^2+NO^2=OA^2$

On putting the values, we get;

$8^2+x^2=r^2\ \ \ \ equation\ 1$

Again,  In ΔCMO

$CM^2+OM^2=OC^2$

$since\ ON=x\\\\\therefore OM=14-x$

$6^2+(14-x)^2=r^2\ \ \ \ \ equation\ 2$

Equation 1 = Equation 2 (due to radius)

$8^2+x^2=6^2+(14-x)^2$

Now we solve the equation to get the value of 'x'.

$64+x^2=36+196+x^2-28x\\\\28x+x^2-x^2=232-64\\\\28x=168\\\\x=\frac{168}{28}=6$

Now putting the value of 'x' in equation 1, we get;

$64+36=r^2\\\\r^2=100\\\\r=\sqrt{100}=10\ cm$

Hence The radius of the circle is 10 cm.