Question Video: Determining the Diameter of a Circle Mathematics

Given that ๐ด๐ต = 14 cm, determine the circleโ€™s diameter.

04:13

Video Transcript

Given that ๐ด๐ต equals 14 centimeters, determine the circleโ€™s diameter.

We should observe that this line segment ๐ด๐ต 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 letโ€™s use some of the information that weโ€™re given, especially the fact that we have a 90-degree angle here at angle ๐‘€๐ถ๐ด.

The fact that we know that ๐ด๐ต is a chord and this angle ๐‘€๐ถ๐ด is 90 degrees means we can use a very important property. This property tells us that if we have a circle with center ๐ด containing a chord line segment ๐ต๐ถ, then the straight line that passes through ๐ด and is perpendicular to line segment ๐ต๐ถ also bisects line segment ๐ต๐ถ. 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 ๐ด๐ต and is perpendicular to it, then that means that ๐ด๐ต is bisected. Therefore, the line segment ๐ด๐ถ is equal in length to the line segment ๐ต๐ถ. And so, if we wanted to work out the length of line segment ๐ด๐ถ, it would be half of the line segment ๐ด๐ต. Half of 14 centimeters is seven centimeters.

Now, letโ€™s consider the triangle ๐‘€๐ถ๐ด, 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 ๐‘€๐ด is useful. This line segment is, in fact, a radius of the circle because itโ€™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, letโ€™s define this line segment ๐ด๐‘€, the radius of the circle, to be ๐‘ฅ 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œƒ degrees, we have the three trigonometric ratios. In this problem, the length of ๐‘ฅ 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 donโ€™t know it. And we donโ€™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, thatโ€™s the opposite side, over the hypotenuse, which we defined as ๐‘ฅ. 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 ๐‘ฅ, giving us that half ๐‘ฅ is equal to seven. And then when we multiply through by two, that gives us that ๐‘ฅ is equal to 14. This means that the length of the line segment ๐‘€๐ด is 14 centimeters.

But of course we havenโ€™t quite finished. Remember that we need to work out the diameter of this circle. Remember, weโ€™ve established that ๐ด๐‘€ must be a radius of the circle. So, the diameter will be double that length, so thatโ€™s 14 times two. This gives us an answer that the diameter of the circle is 28 centimeters.

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