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The line segment π΄π΅ is a chord in circle π whose radius is 25.5 centimeters.
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If the length of π΄π΅ is 40.8 centimeters, what is the length of the line segment π·πΈ?
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The line segment π·πΈ is the segment highlighted in orange.
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Itβs the line that joins the point π·, which is inside the circle, to the point πΈ on the circle circumference.
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Notice that this is a segment of the line ππΈ, which connects the center of the circle to the point on the circumference, and is therefore a radius of the circle.
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The length of the line segment ππΈ then is 25.5 centimeters as this is the radius of the circle given in the question.
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We could therefore work out the length of the line segment π·πΈ by subtracting the length of ππ· from ππΈ.
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So we need to consider how we could work out the length of ππ·.
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Letβs join in a line connecting the points π and π΅ together.
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π is the center of the circle.
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And π΅ is a point on the circumference.
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Itβs the endpoint of our chord π΄π΅.
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And therefore, ππ΅ is the radius of the circle, which means its length is 25.5 centimeters as given in the question.
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We now have a right-angle triangle, triangle ππ·π΅.
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And we know the length of the hypotenuse.
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Weβre also told in the question that the length of the chord π΄π΅ is 40.8 centimeters.
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And we can use this information to work out the length of the line segment π΅π·.
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We know that the perpendicular bisector of a chord passes through the center of the circle.
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Now, the line ππ· or ππΈ does pass through the center of the circle.
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It passes through the point π.
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And from the diagram, we know that it meets the chord π΄π΅ at right angles.
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Therefore, by the converse of this statement, it must be the perpendicular bisector of the chord π΄π΅.
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So this tells us that the lengths of π·π΅ and π·π΄ are equal because the chord has been bisected.
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We can therefore find each of these lengths by halving the length of the chord π΄π΅.
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40.8 divided by two is 20.4.
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We now know two of the lengths in our right-angle triangle.
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So we can apply the Pythagorean theorem in order to work out the third length ππ·, which is one of the two shorter sides.
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The Pythagorean theorem tells us that, in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
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In our triangle, that means that ππ· squared plus 20.4 squared is equal to 25.5 squared.
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And we can solve this equation to find the length of ππ·.
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20.4 squared is 416.16.
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And 25.5 squared is 650.25.
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Subtracting 416.16 from each side of this equation gives that ππ· squared is equal to 234.09.
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To find the value of ππ· then, we need to take the square root of each side of this equation, taking only the positive square root as ππ· is a length.
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We have that ππ· is equal to the square root of 234.09, which is 15.3.
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So now that weβve calculated the length of ππ·, we can return to our calculation for the length of π·πΈ, which we said we were going to work out by subtracting the length of ππ· from ππΈ, which was the radius of the circle.
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Substituting the lengths for ππΈ, 25.5, and ππ·, 15.3, we have that the length of π·πΈ is equal to 25.5 minus 15.3, which gives 10.2.
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The units for this are centimeters because the radius of the circle was also given in centimeters.
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So by recalling that the perpendicular bisector of a chord passes through the center of the circle and then applying the Pythagorean theorem, we found that the length of the line segment π·πΈ is 10.2 centimeters.