Problem Solving

If a circle passes through points (1,2), (2,5), and (5,4), what is the diameter of the circle?

√18
√20
√22
√26
√30

OA: B

richachampion wrote:If a circle passes through points (1,2), (2,5), and (5,4), what is the diameter of the circle?

√18
√20
√22
√26
√30

Plot and connect (1,2), (2,5) and (5,4):


Notice that the angle formed by the three points is a RIGHT ANGLE.
Slope formed by (1,2) and (2,5) = (5-2)/(2-1) = 3.
Slope formed by (2,5) and (5,4) = (4-5)/(5-2) = -1/3.
Since the slopes are negative reciprocals, the two line segments are perpendicular and thus form a right angle.

This right angle constitutes an INSCRIBED ANGLE in the circle:


An inscribed right angle in a circle intercepts the DIAMETER:


To determine the length of the diameter, draw a right triangle, as follows:

In the resulting right triange:
4² + 2² = d²
20 = d²
d = √20.

The correct answer is B.