𝐴𝐵𝐶 is an equilateral triangle with a side length of four centimeters. A circle of radius four centimeters is drawn at point 𝐴. Is the line segment 𝐴𝐶 the radius, chord, or diameter of the circle?
Let’s start this by sketching out this triangle 𝐴𝐵𝐶. We’re told it’s an equilateral triangle and the side length is four centimeters. Because it’s an equilateral triangle, that means that all sides will be the same length. It doesn’t matter which of the vertices we label with 𝐴, 𝐵, or 𝐶 so long as they’re written in order, either clockwise or counterclockwise.
The next thing we’re told is that there is a circle of radius four centimeters, which is drawn at point 𝐴. Don’t forget that the radius is a straight line extending from the center to the circumference or the outside of the circle. With the center of this circle at point 𝐴, then we could draw a circle like this. Notice that there will be an infinite number of radii extending from the center at 𝐴 to the circumference of the circle. But these would all be of length four centimeters.
The final part of the question then asks us if the line segment 𝐴𝐶 is the radius, chord, or diameter of the circle. Let’s highlight this line segment and then recall what the diameter and chord of a circle is.
The diameter of a circle is a line segment passing through the center of a circle and joining two points on the circumference. Notice that a diameter is double the length of the radius. Next then, a chord, it’s defined as a line segment joining two distinct points on the circumference. When we consider the meanings of these three words, we can then say that the line segment 𝐴𝐶 must be a radius. But let’s be sure of why.
Firstly, we can remember that 𝐴𝐶 is part of this equilateral triangle, which has a side length of four centimeters. Next, we know that we have this circle, which has a center at 𝐴 and an infinite number of radii which extend four centimeters. And so we can say that the line segment 𝐴𝐶 is a radius of the circle.
We can say then that it’s not a diameter. If it was, it would be of length eight centimeters. We can also say that the line segment 𝐴𝐶 is not a chord because it doesn’t join two points on the circumference. It is a line segment from the center of the circle to the circumference. Therefore, we can give the answer that the line segment 𝐴𝐶 is a radius of the circle.