Question

Given : O is the centre of the circle. AB and CD are chords. OM is perpendicular to AB and ON perpendicular to CD. AB =30 units and OM =20 units.
calculate :
the length of the the radius OB and the length of CD if ON =15 units

Answer 1

First, let's denote the radius OB as r and the length of CD as x.

Since OM is perpendicular to AB, we can use the Pythagorean theorem to find r:
r^2 = OM^2 + AB^2
r^2 = 20^2 + 30^2
r^2 = 400 + 900
r^2 = 1300
r = sqrt(1300)
r ≈ 36.06 units

Now, let's consider the triangle ONC:
Since ON is perpendicular to CD, triangle ONC is a right-angled triangle.
Applying the Pythagoras Theorem -
OC^2 = ON^2 + NC^2
OC^2 = 15^2 + x^2
OC^2 = 225 + x^2

In triangle OMC, we know OM = 20 units while OC = OB - r = 36.06 - r
Now, using Pythagoras Theorem again,
OC^2 = OM^2 + MC^2
(36.06 - r)^2 = 20^2 + MC^2
(36.06 - 36.06)^2 = 20^2 + MC^2
MC^2 = 400
MC = 20 units

Combining both equations involving MC,
20^2 = 225 + x^2
400 = 225 + x^2
x^2 = 400 - 225
x^2 = 175
x = sqrt(175)
x ≈ 13.23 units

Therefore, the length of the radius OB is approximately 36.06 units and the length of CD is approximately 13.23 units.