Question

If AB = 5 cm, BC = 12 cm, and AB is perpendicular to BC, then find the radius of the circle passing through the points A, B, and C.

Answer 1

The radius of the circle passing through points A, B, and C, with AB perpendicular to BC, is found using the Pythagorean theorem. The diameter of the circle is the hypotenuse AC of the right-angled triangle ABC. The radius is half of the hypotenuse, which is 6.5 cm.

Step-by-step explanation:

The student is tasked with finding the radius of a circle passing through three points A, B, and C, where AB is perpendicular to BC, and the lengths AB and BC are given. This is a geometry problem that involves constructing a circle through these three points, which will form a right-angled triangle. A right-angled triangle inscribed in a circle implies that the hypotenuse AC of the triangle will be the diameter of the circle.

Step-by-step Solution:

1. First, we recognize that triangle ABC is a right-angled triangle (because AB is perpendicular to BC).
2. By applying the Pythagorean theorem to triangle ABC, we can find AC: AC^2 = AB^2 + BC^2.
3. After calculating AC, we can determine the radius of the circle, since it is half of AC (the diameter).

Therefore, AC=√(5^2 + 12^2)cm = √(25 + 144)cm = √169cm = 13cm. The radius (r) of the circle is half of AC, so r = AC/2 = 13/2 = 6.5 cm.