𝑀 is a circle with radius 24 centimeters. A chord is drawn whose central angle is 62 degrees. Find the length of the chord, giving the answer to the nearest centimeter.
Here’s our circle, 𝑀, the radius of circle 𝑀, here’s the chord whose central angle is 62 degrees. This chord creates an isosceles triangle. We know that the third side of this triangle is a radius of circle 𝑀 and measures 24 centimeters. We want to know the length of this third side. We know a side, an angle, and a side. And that means we can use the law of cosine to find the missing length.
The law of cosine tells us 𝑐 squared, a side squared, equals 𝑎 squared plus 𝑏 squared, so the other two sides both squared, minus two times 𝑎 times 𝑏 times the cosine of 𝑐. That means the cosine of the angle opposite side 𝑐. We’ll label our chord as side 𝑐, one radius as side 𝑎, and the other radius as side 𝑏. We don’t know 𝑐, so we keep it as the variable 𝑐 squared equals 24 squared plus 24 squared minus two times 24 times 24 times the cosine of 62 degrees.
24 squared equals 576. I noticed that we’re multiplying 24 times 24 and the second half of the equation. That’s the same thing as 24 squared. It’s equal to 576 and we’ll just bring everything else down. Next, two times 576 equals 1152. Bring down the cosine of 62 degrees, and then we’ll go ahead and add 576 plus 576. That equals 1152 as well.
Our next step is to multiply 1152 times the cosine of 62. If we round it to the nearest hundredth, we get 540.83. If you use your calculator and you did not get this value, you need to make sure that your calculator is using degrees and not radians. Bring down the rest of the equation, 1152 minus 540.83 equals 611.17. We can’t forget that this is our length squared. That’s our 𝑐 squared.
To find 𝑐, we need to take the square root of both sides of the equation. The square root of 611.17 is 24.72. We want to round this value to the nearest centimeter. That’s the nearest whole number. There’s a seven in the tenths place. And that means that we round our ones place up from four to five. Everything to the left stays the same. And 𝑐 equals 25. But 25 what? 25 centimeters. The chord created inside the circle from the angle 62 degrees measures 25 centimeters.