Question

AB = 16 cm, OM = 15 cm, ON = 8 cm

CD =

Options

• 15 cm

• 30 cm

• 13 cm

• 26 cm

Answer 1

30 cm

Explanation:

To find CD, let’s visualize the problem:

• AB = 16 cm is the length of the chord. 
• OM = 15 cm is the perpendicular distance from the center O to the midpoint M of the chord AB.
• ON = 8 cm is the distance from O to a point N on the radius of the circle.

Since OM is the perpendicular bisector of AB, it divides AB into two equal halves.

Therefore, half of AB is:

`(AB)/2 = 16/2 = 8  cm`

Now, we need to find the length of CD, which is the distance between C and D, where C and D are points on the circumference of the circle. This distance is the chord of a different segment of the circle and we need to use the Pythagorean theorem to find it.

From the right triangle formed by the radius the line ON and the distance from the center to the chord, we use the relation:

`r^2 = ON^2 + ((CD)/2)^2`

We know the radius r = 17 cm from the previous calculation and ON = 8 cm.

Thus:

`17^2 = 8^2 + ((CD)/2)^2`

`289 = 64 + ((CD)/2)^2`

`289 - 64 = ((CD)/2)^2`

`225 = ((CD)/2)^2`

Taking the square root of both sides:

`(CD)/2 = 15`

Therefore:

CD = 30 cm