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Circle π has the radius of 12 centimeters where the length of πΆπ΅ is 16 centimeters.
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Find the length of arc πΆπ΅ giving the answer to two decimal places.
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First, letβs put all the information weβre given onto the diagram.
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Weβre told the circle has a radius of 12 centimeters.
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So the lines ππΆ and ππ΅ are each 12 centimeters.
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Weβre also told that the length of the line segment πΆπ΅ is 16 centimeters.
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What weβre looking to calculate in this question is the length of the arc πΆπ΅, the part that Iβve marked in pink.
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So in order to do this, we need to know the size of the central angle, the angle thatβs been marked as π in the diagram.
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Now we havenβt been given angle π, so weβre going to need to calculate it from the other information in the question.
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What you notice is that the π lies inside the triangle ππ΅πΆ, and we know the lengths of all three sides.
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Theyβre 12 centimeters, 12 centimeters and 16 centimeters.
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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.
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So the law of cosines for an angle, using the letters in this question, tells us that cos of π is equal to π΅π squared plus πΆπ squared minus π΅πΆ squared over two multiplied by π΅π multiplied by πΆπ.
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So letβs substitute the values for each of these lengths.
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This tells us that cos of π is equal to 12 squared plus 12 squared minus 16 squared over two multiplied by 12 multiplied by 12.
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Evaluating each of these tells us that cos of π is equal to 32 over 288.
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Now to find the value of π, we need to use the inverse cosine function.
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π is equal to cosine inverse of 32 over 288.
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Evaluating this on my calculator tells me that π is equal to 83.62062 dot dot dot.
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Now Iβm going to keep this value on my calculator screen because I need to use it in the next stage of my calculation, and I donβt want to make my answer inaccurate by introducing any rounding errors.
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So the next stage of this question is we need to actually calculate the length of this arc πΆπ΅.
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So the arc length can be found by finding the circumference of the full circle two ππ and then multiply it by the portion of the circle that we have.
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So thatβs π over 360.
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So thatβs 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.
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I have 83.62062 divided by 360, and then multiplied by two times π times the radius of circle which is 12.
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So if I evaluate all of this on my calculator, I have a value of 17.513463.
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And if I return to the question, it asks me to give my answer to two decimal places.
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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 πΆπ΅ is 17.51 centimeters.