Finding the Number of Circles Passing Through Three Points

The question asks how many circles can pass through three specific points A, B, and C, given the distances between them: AB = 9 cm, BC = 11 cm, and AC = 20 cm.

A fundamental concept in geometry is that three non-collinear points define a unique circle. This circle is known as the circumcircle of the triangle formed by the three points. However, if the three points are collinear (lie on the same straight line), they cannot form a triangle, and a standard circle cannot pass through all three distinct points simultaneously.

Checking for Collinearity of Points A, B, C

To determine if points A, B, and C are collinear, we check the relationship between the given distances. For three points to be collinear, the sum of the lengths of the two shorter segments must be equal to the length of the longest segment. The given lengths are:

• AB = 9 cm
• BC = 11 cm
• AC = 20 cm

Let's check if the sum of the two shorter lengths equals the longest length:

\( \text{AB} + \text{BC} = 9 \text{ cm} + 11 \text{ cm} = 20 \text{ cm} \)

We compare this sum to the length of the longest segment, AC:

\( \text{AC} = 20 \text{ cm} \)

Since \( \text{AB} + \text{BC} = \text{AC} \) (\( 20 \text{ cm} = 20 \text{ cm} \)), the points A, B, and C are collinear. Point B lies between points A and C.

Circles Through Collinear Points

As established, three distinct collinear points cannot lie on a single standard circle. A straight line can intersect a circle at most at two distinct points. Since A, B, and C are three distinct points on a straight line, no circle can pass through all three of them.

Therefore, the number of circles passing through points A, B, and C is zero.

Summary of Analysis

In this specific case, points A, B, and C are collinear because the sum of the lengths of two segments (AB + BC) equals the length of the third segment (AC). Consequently, no circle can pass through all three points.

Conclusion on the Number of Circles

Based on the collinearity of points A, B, and C, the number of circles that can pass through all three points is 0.

Revision Table: Geometry of Points and Circles

Additional Information: Circumcircles and Degeneracy

The concept of a circumcircle is central when considering circles passing through points. For any triangle, there is exactly one circumcircle. Its center is the circumcenter, which is the intersection of the perpendicular bisectors of the triangle's sides. The radius is the distance from the circumcenter to any vertex.

When points A, B, and C are collinear, they do not form a triangle in the traditional sense. The "triangle" is degenerate, meaning it has zero area and its vertices lie on a single line segment. In this degenerate case, the perpendicular bisectors of the "sides" (AB, BC, AC) would all be parallel lines (perpendicular to the line containing A, B, C), and thus would not intersect at a single point. This geometric property further illustrates why no single circle can be defined by three distinct collinear points.

It's important to distinguish this from cases involving fewer points: two distinct points define infinitely many circles (their centers lie on the perpendicular bisector of the segment connecting them), and a single point also lies on infinitely many circles.