Given : circle with centre O and chord AB. OC Is perpendicular to AB, cutting AB at D, with C on the circumference. OB=13 units and AB=24 units. Calculate the length of CD.

Answer 1

First, let's find the radius of the circle using the Pythagorean Theorem.

Since OC is perpendicular to AB, we can use triangle ODC to find the radius.

OD^2 + CD^2 = OC^2

Since OC is the radius of the circle, let's denote it as r.

(13 - r)^2 + CD^2 = r^2

Expanding the left side:

169 - 26r + r^2 + CD^2 = r^2

CD^2 = 169 - 26r

Now, let's use the fact that OC is perpendicular to AB and form right triangle ABC.

Using Pythagorean Theorem in triangle ABC:

AC^2 + BC^2 = AB^2

AC^2 + 169 = 576

AC^2 = 407

Now, use Pythagorean Theorem in triangle ACD:

AC^2 = AD^2 + CD^2

407 = AD^2 + CD^2

Since we know that AC = 13 + r, we can substitute:

(13 + r)^2 = AD^2 + CD^2
169 + 26r + r^2 = AD^2 + CD^2

Substitute the value of AC^2 into the equation:

AD^2 + CD^2 = 407

Using this equation and the previous equation for CD^2, we can solve for CD.

169 + 26r + r^2 = 407
26r + r^2 = 238
r^2 + 26r - 238 = 0
(r + 34)(r - 7) = 0

Since r cannot be negative, r = 7 units.

Therefore, CD^2 = 169 - 26(7) = 169 - 182 = -13

Since the length of a segment cannot be negative, the calculation might have an error.