Given that 𝐴𝑀 equals 50 centimeters, find the length of 𝐵𝐶.
So we have a diagram of a circle with center 𝑀. And we’re asked to calculate the length of 𝐵𝐶, which is a chord of the circle. We’re also told that the length of the line 𝐴𝑀 is 50 centimeters.
If we consider this line 𝐴𝑀, we can see that it is in fact a radius of the circle as its end points are the center of the circle 𝑀 and a point on the circumference 𝐴. The line 𝐴𝐵 is a diameter of the circle as its two end points are both points on the circumference and it passes through the center.
The relationship between the diameter and radius of a circle is that the diameter is twice as long as the radius. So if 𝐴𝑀 is 50 centimeters, then the full length of 𝐴𝐵 is twice this. It’s 100 centimeters.
We’re looking to calculate the length of the chord 𝐵𝐶, which is a side in the triangle 𝐴𝐵𝐶, in which we know one other side, 100 centimeters, and one angle, 30 degrees. Do we know any other information about this triangle?
Well, in fact, we do. The line 𝐴𝐵 as we’ve already discussed is the diameter of the circle dividing it up into two semicircles. We know that the angle inscribed in a semicircle is 90 degrees. An inscribed angle is one which has its vertex on the circle, so that’s the point 𝐶, and sides that contain chords of the circle, so that’s the chords 𝐵𝐶 and 𝐴𝐶.
Therefore, we know that the angle 𝐵𝐶𝐴 is a right angle, 90 degrees. So now we have a right-angled triangle in which we know the length of one side, the size of one other angle, and are looking to calculate the length of a second side. We can therefore apply trigonometry to this problem.
First, we need to decide which trigonometric ratio to apply. And in order to do this, we need to begin by labelling the three sides of the triangle in relation to the angle of 30 degrees. 𝐵𝐶 is the opposite side. 𝐴𝐶 is the adjacent. And 𝐴𝐵 is the hypotenuse.
We can recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios to use. The side we want to calculate is the opposite. And the side we know is the hypotenuse. O and H appear together in the SOH part of this acronym, which tells us that it’s the sine ratio that we need to use in this question.
Let’s recall its definition. The sin of an angle 𝜃 is equal to the opposite side divided by the hypotenuse. In this question, the angle 𝜃 is 30 degrees. The opposite is the side we want to calculate, 𝐵𝐶. And the hypotenuse is 100. So we have the equation sin of 30 degrees is equal to 𝐵𝐶 over 100.
We’d like to solve this equation to find the length of 𝐵𝐶. Let’s begin by multiplying both sides by 100. I’ve also written the two sides of the equation the other way round at this stage so that 𝐵𝐶 is on the left-hand side. We have 𝐵𝐶 is equal to 100 sin 30 degrees.
Now we can in fact answer this question without using a calculator because 30 degrees is one of the key angles for which we need to know the sine, cosine, and tangent ratios of by heart.
The value of sin of 30 degrees is in fact just equal to a half. Therefore, 𝐵𝐶 is equal to 100 multiplied by a half. And our answer to the problem is that the length of 𝐵𝐶 is 50 centimeters. We answered this question by applying trigonometry in a right-angled triangle. In order to do this, we needed to recall the key fact that the angle inscribed in a semicircle is 90 degrees.