### Video Transcript

Given that ๐ด๐ equals 50 centimeters, find the length of ๐ต๐ถ.

So we have a diagram of a circle with center ๐. And weโre asked to calculate the length of ๐ต๐ถ, which is a chord of the circle. Weโre also told that the length of the line ๐ด๐ is 50 centimeters.

If we consider this line ๐ด๐, we can see that it is in fact a radius of the circle as its end points are the center of the circle ๐ and a point on the circumference ๐ด. The line ๐ด๐ต 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 ๐ด๐ is 50 centimeters, then the full length of ๐ด๐ต is twice this. Itโs 100 centimeters.

Weโre looking to calculate the length of the chord ๐ต๐ถ, which is a side in the triangle ๐ด๐ต๐ถ, 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 ๐ด๐ต as weโve 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 thatโs the point ๐ถ, and sides that contain chords of the circle, so thatโs the chords ๐ต๐ถ and ๐ด๐ถ.

Therefore, we know that the angle ๐ต๐ถ๐ด 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. ๐ต๐ถ is the opposite side. ๐ด๐ถ is the adjacent. And ๐ด๐ต 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 itโs the sine ratio that we need to use in this question.

Letโs recall its definition. The sin of an angle ๐ is equal to the opposite side divided by the hypotenuse. In this question, the angle ๐ is 30 degrees. The opposite is the side we want to calculate, ๐ต๐ถ. And the hypotenuse is 100. So we have the equation sin of 30 degrees is equal to ๐ต๐ถ over 100.

Weโd like to solve this equation to find the length of ๐ต๐ถ. Letโs begin by multiplying both sides by 100. Iโve also written the two sides of the equation the other way round at this stage so that ๐ต๐ถ is on the left-hand side. We have ๐ต๐ถ 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, ๐ต๐ถ is equal to 100 multiplied by a half. And our answer to the problem is that the length of ๐ต๐ถ 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.