WEBVTT
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๐ is a circle with radius 24 centimeters.
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A chord is drawn whose central angle is 62 degrees.
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Find the length of the chord, giving the answer to the nearest centimeter.
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Hereโs our circle, ๐, the radius of circle ๐, hereโs the chord whose central angle is 62 degrees.
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This chord creates an isosceles triangle.
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We know that the third side of this triangle is a radius of circle ๐ and measures 24 centimeters.
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We want to know the length of this third side.
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We know a side, an angle, and a side.
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And that means we can use the law of cosine to find the missing length.
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The law of cosine tells us ๐ squared, a side squared, equals ๐ squared plus ๐ squared, so the other two sides both squared, minus two times ๐ times ๐ times the cosine of ๐.
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That means the cosine of the angle opposite side ๐.
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Weโll label our chord as side ๐, one radius as side ๐, and the other radius as side ๐.
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We donโt know ๐, so we keep it as the variable ๐ squared equals 24 squared plus 24 squared minus two times 24 times 24 times the cosine of 62 degrees.
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24 squared equals 576.
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I noticed that weโre multiplying 24 times 24 and the second half of the equation.
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Thatโs the same thing as 24 squared.
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Itโs equal to 576 and weโll just bring everything else down.
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Next, two times 576 equals 1152.
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Bring down the cosine of 62 degrees, and then weโll go ahead and add 576 plus 576.
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That equals 1152 as well.
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Our next step is to multiply 1152 times the cosine of 62.
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If we round it to the nearest hundredth, we get 540.83.
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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.
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Bring down the rest of the equation, 1152 minus 540.83 equals 611.17.
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We canโt forget that this is our length squared.
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Thatโs our ๐ squared.
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To find ๐, we need to take the square root of both sides of the equation.
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The square root of 611.17 is 24.72.
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We want to round this value to the nearest centimeter.
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Thatโs the nearest whole number.
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Thereโs a seven in the tenths place.
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And that means that we round our ones place up from four to five.
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Everything to the left stays the same.
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And ๐ equals 25.
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But 25 what?
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25 centimeters.
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The chord created inside the circle from the angle 62 degrees measures 25 centimeters.