Question

Find the value of x. Assume that the line segment labeled 6 is tangent to the circle.

A. 6.5
B. 2.9
C. 5.5
D. 10.6

Answer 1

We can use the fact that a tangent line to a circle is perpendicular to the radius at the point of tangency. Therefore, we can draw a radius from the center of the circle to the point of tangency on the line segment labeled 6. This creates a right triangle where the radius is the hypotenuse and one leg is x.

Using the Pythagorean theorem, we have:

$r^2 = x^2 + 6^2$

Substituting the given value of $r=11$:

$11^2 = x^2 + 6^2$

Simplifying:

$121 = x^2 + 36$

$x^2 = 121 - 36 = 85$

Taking the square root of both sides (and remembering to consider both the positive and negative roots since $x$ could be on either side of the center of the circle), we get:

$x = \pm \sqrt{85}$

Since the answer choices are all positive numbers, we can discard the negative root and choose:

$x = \sqrt{85}$

Using a calculator, we find that $\sqrt{85} \approx 9.22$, which is closest to answer choice:

$\boxed{\textbf{(D) }10.6}$