In a circle with center O, chord AB has a length of 12. The perpendicular distance from ce… — SAT

In a circle with center $O$ , chord $AB$ has a length of $12$ . The perpendicular distance from center $O$ to chord $AB$ is $8$ . If the area of the circle is $k\pi$ , what is the value of $k$ ?

Answer: 100

Answer

100

A perpendicular from the center of a circle to a chord bisects that chord. For a chord of length

12

, the perpendicular split creates two segments of length

6

. Drawing a radius from the center to one of the chord's endpoints forms a right triangle with legs of

6

and

8

. By the Pythagorean theorem, the hypotenuse (which is the radius

r

) satisfies

r^2 = 6^2 + 8^2 = 100

. The area of the circle is

\pi r^2 = 100\pi

. Thus, the coefficient

k

100

Step-by-Step Solution

Determine the length of half of the chord.

A line segment drawn perpendicular from the center of a circle to a chord bisects the chord. Therefore, the distance from the midpoint of the chord to either endpoint is

12 / 2 = 6

Use the Pythagorean theorem to calculate the square of the radius.

r^2 = 100

The radius, half of the chord, and the perpendicular distance form a right-angled triangle. According to the Pythagorean theorem, the hypotenuse squared (

r^2

) is the sum of the squares of the legs:

r^2 = 6^2 + 8^2 = 36 + 64 = 100

Calculate the area of the circle in terms of

\pi

and identify the value of

k

k = 100

The formula for the area of a circle is

\pi r^2

. Since

r^2 = 100

, the area is

100\pi

. Comparing this to

k\pi

, we find that

k = 100

Key Concept

Perpendicular bisector of a circle chord and right triangle properties

Estimated Time:1m 30s

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