Question

Triangle \(A B C\) is contained within a circle with center \(C .\) Points \(A\) and \(B\) lie on the circle. If the area of circle \(C\) is \(25 \pi,\) and the measure of angle \(A C B\) is \(60^{\circ},\) which of the following are possible lengths for side \(A B\) of triangle \(A B C ?\) Indicate all such lengths. a. 3 b. 4 c. 5 d. 6 e. 7

Short answer

The possible length for side AB of triangle ABC is c. 5

Step-by-step solution

Calculate radius of the circle

The area of a circle is given by the formula \(A = \pi r^2\), where A is the area and r the radius of the circle. Given that the area A is \(25 \pi\), the radius can be calculated by rearranging the expression as: \(r = \sqrt{A/\pi}\). Substituting \(A = 25 \pi\), the radius \(r\) is found to be 5.

Calculate diameter of the circle

The diameter of a circle is twice the radius. Since the radius \(r\) is 5, the diameter is \(2 \times 5 = 10\). In triangle ABC, side ACB is the diameter of the circle.

Determine lengths of side AB

Since triangle ACB is an isosceles triangle (\(A C = C B = 5\) and angle \(ACB = 60^{\circ}\)), side AB, which is opposite the \(60^{\circ}\) angle, forms the base of an equilateral triangle (since all angles are \(60^{\circ}\)). Therefore, AB is also equal to the radius which is 5. Now, all options can be checked and it is found that only option c. 5 is possible for side AB

Key concepts

Circle Area

Understanding the area of a circle is a foundational concept in geometry. The formula for calculating the area of a circle is given by

• \(A = \pi r^2\)

where \(A\) is the area and \(r\) is the radius of the circle.
This formula essentially tells you how much space is enclosed within the boundary of a circle. In our exercise, the area of the circle is provided as \(25 \pi\). To find the radius, you would rearrange the formula as \(r = \sqrt{\frac{A}{\pi}}\).
By substituting \(A = 25 \pi\), the extraction leads to a radius of 5 units. This radius is pivotal since it helps in defining additional circle properties, such as diameter and circumference.

Triangle Properties

Triangles come with a wide array of intriguing properties, and each type holds its own unique features. For a triangle inscribed in a circle (a circumscribed triangle), the diameter of the circle plays a vital part.
The circle’s diameter doubles the radius, so when its diameter acts as one side of a triangle, special properties arise.
In general, triangles have basic properties, including:

• Sum of interior angles equals \(180^{\circ}\)
• Length of any two sides must be greater than the third side
• Types of triangles: scalene (no equal sides), isosceles (two equal sides), and equilateral (all equal sides)

If a triangle is inscribed in a circle where one side is the diameter, it must necessarily be a right triangle (by Thales' theorem), unless of course, all angles are equilateral such as in our case. Understanding these properties intricately is crucial for solving triangular problems accurately.

Isosceles Triangle

An isosceles triangle is characterized by having two sides of equal length. These two sides are known as legs, and the angle between them is often a point of interest.
The base of the isosceles triangle is the side that differs in length and does not lie between equal angles. Equal base angles are another key feature of these triangles.
Applying this understanding to triangle \(ABC\), since \(AC = CB = 5\), it is an isosceles triangle.
As angle \(ACB = 60^{\circ}\), all angles in a triangle sum to \(180^{\circ}\). Therefore, if \(ACB\) is \(60^{\circ}\), then each base angle must also equal \(60^{\circ}\), forming an equilateral triangle. Thus, all sides, including \(AB\), measure 5 units.
Breaking down these properties meticulously allows for a clearer understanding of different rules governing isosceles triangles and their application in geometrical problems.