Given that 𝐴𝐵 equals 14 centimeters, determine the circle’s diameter.
We should observe that this line segment 𝐴𝐵 is, in fact, a chord of the circle. A chord of a circle is a line segment joining two distinct points on the circumference. In this problem, we need to work out the diameter of the circle. We should remember that a diameter is a line which joins two points on the circumference but which passes through the center of a circle. It might be difficult to work out just exactly how we are going to find the length of the diameter in this circle. But let’s use some of the information that we’re given, especially the fact that we have a 90-degree angle here at angle 𝑀𝐶𝐴.
The fact that we know that 𝐴𝐵 is a chord and this angle 𝑀𝐶𝐴 is 90 degrees means we can use a very important property. This property tells us that if we have a circle with center 𝐴 containing a chord line segment 𝐵𝐶, then the straight line that passes through 𝐴 and is perpendicular to line segment 𝐵𝐶 also bisects line segment 𝐵𝐶. We can even adapt this information to fit the lettering in this circle. Because we know that there is a line from the center 𝑀 which passes through the chord 𝐴𝐵 and is perpendicular to it, then that means that 𝐴𝐵 is bisected. Therefore, the line segment 𝐴𝐶 is equal in length to the line segment 𝐵𝐶. And so, if we wanted to work out the length of line segment 𝐴𝐶, it would be half of the line segment 𝐴𝐵. Half of 14 centimeters is seven centimeters.
Now, let’s consider the triangle 𝑀𝐶𝐴, and we know, of course, that this will be a right triangle. When we have a right triangle, we may be able to apply the Pythagorean theorem or some trigonometry. We might even start to realize that this line segment 𝑀𝐴 is useful. This line segment is, in fact, a radius of the circle because it’s a line segment going from the center of the circle to a point on the circumference. Once we know the value of the radius, then we can easily work out the diameter of the circle. So, let’s define this line segment 𝐴𝑀, the radius of the circle, to be 𝑥 centimeters.
The fact that we have a side and a side that we wish to calculate along with an angle means that we should apply some trigonometry. For an included angle of 𝜃 degrees, we have the three trigonometric ratios. In this problem, the length of 𝑥 centimeters represents the hypotenuse in the right triangle. The length of seven centimeters is opposite the angle of 30 degrees. The third side is adjacent to the angle of 30 degrees, but we don’t know it. And we don’t wish to calculate it, so we can discount it.
The ratio which involves the opposite side and the hypotenuse is that of the sine ratio. We can apply this ratio with sin 30 degrees is equal to seven, that’s the opposite side, over the hypotenuse, which we defined as 𝑥. It can be useful at this point to remember that sin of 30 degrees is equal to one-half. We can then multiply both sides of this equation by 𝑥, giving us that half 𝑥 is equal to seven. And then when we multiply through by two, that gives us that 𝑥 is equal to 14. This means that the length of the line segment 𝑀𝐴 is 14 centimeters.
But of course we haven’t quite finished. Remember that we need to work out the diameter of this circle. Remember, we’ve established that 𝐴𝑀 must be a radius of the circle. So, the diameter will be double that length, so that’s 14 times two. This gives us an answer that the diameter of the circle is 28 centimeters.