The circle in the figure has a radius of r and center at C . The distance from A to B is x . If and , find x . Round the answer to the nearest whole number.
A)10
B)17
C)13
D)12
E)15

Question

The circle in the figure has a radius of r and center at C . The distance from A to B is x .  ​ If   and  , find x . Round the answer to the nearest whole number. ​
A)10 
B)17 
C)13 
D)12 
E)15

Quizplus · Accepted Answer

Draw radii from the center C to points A and B, creating right triangles. Let's call the distance from C to point D on segment AB h. Since the radius of the circle is r, we know that CD = r.

Using the Pythagorean theorem in triangle ACD, we have:
AC^2 + CD^2 = AD^2
AC^2 + r^2 = (x/2)^2
Simplifying, we get:
AC^2 = (x^2/4) - r^2

Using the Pythagorean theorem in triangle BCD, we have:
BC^2 + CD^2 = BD^2
BC^2 + r^2 = (x/2)^2
Simplifying, we get:
BC^2 = (x^2/4) - r^2

Adding the two equations together, we get:
AC^2 + BC^2 = (x^2/2) - 2r^2

Substituting AC^2 and BC^2 in terms of x and r, we get:
(x^2/4) - r^2 + (x^2/4) - r^2 = (x^2/2) - 2r^2
Simplifying, we get:
x^2 = 8r^2
Taking the square root of both sides, we get:
x = sqrt(8) * r
Asking for the answer rounded to the nearest whole number, we get:
x = 3.54 * r
Since r is a positive number, x will always be greater than 3.54r. The closest answer choice to this value is 13, so the answer is (C).

The Circle in the Figure Has a Radius of R