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.
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.