Video: Finding the Length of an Arc of a Circle given Its Radius and a Subtended Chord Using the Cosine Rule

Circle ๐‘€ has a radius of 12 cm where the length of ๐ถ๐ต is 16 cm. Find the length of arc ๐ถ๐ต giving the answer to two decimal places.

03:37

Video Transcript

Circle ๐‘€ has the radius of 12 centimeters where the length of ๐ถ๐ต is 16 centimeters. Find the length of arc ๐ถ๐ต giving the answer to two decimal places.

First, letโ€™s put all the information weโ€™re given onto the diagram. Weโ€™re told the circle has a radius of 12 centimeters. So the lines ๐‘€๐ถ and ๐‘€๐ต are each 12 centimeters. Weโ€™re also told that the length of the line segment ๐ถ๐ต is 16 centimeters. What weโ€™re looking to calculate in this question is the length of the arc ๐ถ๐ต, the part that Iโ€™ve marked in pink.

So in order to do this, we need to know the size of the central angle, the angle thatโ€™s been marked as ๐œƒ in the diagram. Now we havenโ€™t been given angle ๐œƒ, so weโ€™re going to need to calculate it from the other information in the question. What you notice is that the ๐œƒ lies inside the triangle ๐‘€๐ต๐ถ, and we know the lengths of all three sides. Theyโ€™re 12 centimeters, 12 centimeters and 16 centimeters.

If we know the lengths of all three sides of a triangle, then we can calculate the size of any of its angles using the law of cosines. So the law of cosines for an angle, using the letters in this question, tells us that cos of ๐œƒ is equal to ๐ต๐‘€ squared plus ๐ถ๐‘€ squared minus ๐ต๐ถ squared over two multiplied by ๐ต๐‘€ multiplied by ๐ถ๐‘€. So letโ€™s substitute the values for each of these lengths. This tells us that cos of ๐œƒ is equal to 12 squared plus 12 squared minus 16 squared over two multiplied by 12 multiplied by 12. Evaluating each of these tells us that cos of ๐œƒ is equal to 32 over 288.

Now to find the value of ๐œƒ, we need to use the inverse cosine function. ๐œƒ is equal to cosine inverse of 32 over 288. Evaluating this on my calculator tells me that ๐œƒ is equal to 83.62062 dot dot dot. Now Iโ€™m going to keep this value on my calculator screen because I need to use it in the next stage of my calculation, and I donโ€™t want to make my answer inaccurate by introducing any rounding errors.

So the next stage of this question is we need to actually calculate the length of this arc ๐ถ๐ต. So the arc length can be found by finding the circumference of the full circle two ๐œ‹๐‘Ÿ and then multiply it by the portion of the circle that we have. So thatโ€™s ๐œƒ over 360. So thatโ€™s why keeping this value on my calculator screen has been really useful cause now I can use it exactly in this stage of the calculation. I have 83.62062 divided by 360, and then multiplied by two times ๐œ‹ times the radius of circle which is 12.

So if I evaluate all of this on my calculator, I have a value of 17.513463. And if I return to the question, it asks me to give my answer to two decimal places. So rounding my answer and including the units for an arc length, which in this case are centimeters, tells me that the length of arc ๐ถ๐ต is 17.51 centimeters.

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