Video Transcript
Given that ๐ด๐ต equals 14 centimeters, determine the circleโs diameter.
We should observe that this line segment ๐ด๐ต 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 letโs use some of the information that weโre given, especially the fact that we have a 90-degree angle here at angle ๐๐ถ๐ด.
The fact that we know that ๐ด๐ต is a chord and this angle ๐๐ถ๐ด is 90 degrees means we can use a very important property. This property tells us that if we have a circle with center ๐ด containing a chord line segment ๐ต๐ถ, then the straight line that passes through ๐ด and is perpendicular to line segment ๐ต๐ถ also bisects line segment ๐ต๐ถ. We can even adapt this information to fit the lettering in this circle. Because we know that there is a line from the center ๐ which passes through the chord ๐ด๐ต and is perpendicular to it, then that means that ๐ด๐ต is bisected. Therefore, the line segment ๐ด๐ถ is equal in length to the line segment ๐ต๐ถ. And so, if we wanted to work out the length of line segment ๐ด๐ถ, it would be half of the line segment ๐ด๐ต. Half of 14 centimeters is seven centimeters.
Now, letโs consider the triangle ๐๐ถ๐ด, 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 ๐๐ด is useful. This line segment is, in fact, a radius of the circle because itโs 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, letโs define this line segment ๐ด๐, the radius of the circle, to be ๐ฅ 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 ๐ degrees, we have the three trigonometric ratios. In this problem, the length of ๐ฅ 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 donโt know it. And we donโt 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, thatโs the opposite side, over the hypotenuse, which we defined as ๐ฅ. 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 ๐ฅ, giving us that half ๐ฅ is equal to seven. And then when we multiply through by two, that gives us that ๐ฅ is equal to 14. This means that the length of the line segment ๐๐ด is 14 centimeters.
But of course we havenโt quite finished. Remember that we need to work out the diameter of this circle. Remember, weโve established that ๐ด๐ must be a radius of the circle. So, the diameter will be double that length, so thatโs 14 times two. This gives us an answer that the diameter of the circle is 28 centimeters.
