Question

given circle x with radius 10 units and chord ab with length 12 units, what is the length of segment cx, which bisects the chord?
a. 16 units
b. 8 units
c. 10 units
d. 6 units

Explanation:

Step1: Recall the property of a chord - bisecting line

A line from the center of a circle that bisects a chord is perpendicular to the chord. So, triangle $XCB$ is a right - triangle, where $XB$ is the radius of the circle and $CB$ is half of the chord $AB$.

Step2: Calculate the length of $CB$

Since chord $AB = 12$ units and $CX$ bisects $AB$, then $CB=\frac{AB}{2}=\frac{12}{2}=6$ units. The radius $XB = 10$ units.

Step3: Apply the Pythagorean theorem

In right - triangle $XCB$, by the Pythagorean theorem $CX=\sqrt{XB^{2}-CB^{2}}$. Substitute $XB = 10$ and $CB = 6$ into the formula: $CX=\sqrt{10^{2}-6^{2}}=\sqrt{100 - 36}=\sqrt{64}=8$ units.

Answer:

B. 8 units