Question

The radius of the circle passing through the points $(2,3)$, $(2,7)$ and $(5,3)$ is:

• A. $5$
• B. $4$
• C. $\frac{5}{2}$
• D. $2$
• E. $\sqrt{5}$

Hint:

Always check if any two points share a coordinate. If two points have the same x-coordinate and another pair has the same y-coordinate, you have a right triangle, and the radius is simply half the distance between the two points that don't share a coordinate.

Answer 1

The Correct Option is C

Concept: A circle passing through three points forms a triangle. If the triangle is a right-angled triangle, the circumcenter lies at the midpoint of the hypotenuse, and the radius is half the length of the hypotenuse.

Step 1: Identify the nature of the triangle formed by the points.
Let the points be $A(2,3)$, $B(2,7)$, and $C(5,3)$.
• $AB$ is a vertical line segment because the x-coordinates are the same ($x=2$). Length $AB = |7 - 3| = 4$.
• $AC$ is a horizontal line segment because the y-coordinates are the same ($y=3$). Length $AC = |5 - 2| = 3$. Since a horizontal line and a vertical line are perpendicular, $\angle BAC = 90^\circ$. Thus, $\triangle ABC$ is a right-angled triangle at $A$.

Step 2: Find the length of the hypotenuse.

The hypotenuse is $BC$. Using the distance formula:
\[ BC = \sqrt{(5 - 2)^2 + (3 - 7)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \]

Step 3: Calculate the radius.
In a right-angled triangle, the radius of the circumcircle ($R$) is half of the hypotenuse: \[ R = \frac{BC}{2} = \frac{5}{2} \]