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Given π΄π equals 200 centimeters and ππΆ equals 120 centimeters, find the length of the line segment π΄π΅.
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So, weβve been given a diagram of a circle and then some internal line segments.
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Weβre also told that π΄π is 200 centimeters and ππΆ is 120 centimeters.
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Now, π΄π wasnβt already drawn on the diagram, so we can go ahead and add that line in and label it with its length.
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We can also include the length of ππΆ.
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Itβs 120 centimeters.
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The length weβre looking to calculate is the length of the line segment π΄π΅, which we can see is a cord of this circle.
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Now, a key point on the diagram is that these little marks have been added to the line segments π΄πΆ and πΆπ΅, which indicate that they are the same length.
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This means that the line segment ππΆ, or ππ· if we go all the way to the circumference of the circle, bisects the line segment π΄π΅ as itβs divided it up into two equal parts.
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This further tells us that the angle where these two lines meet must be a right angle because if a line drawn from the center of a circle to the circumference divides a cord in half, then it will be a perpendicular bisector of that cord.
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So, we know that ππ· is the perpendicular bisector of π΄π΅.
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And by drawing in this right angle, we see that we now have a right triangle π΄ππΆ in which we know two lengths.
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We can, therefore, apply the Pythagorean theorem to find the third length π΄πΆ.
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And as π΄πΆ is half of π΄π΅, weβll be able to double this value to find the total length of π΄π΅.
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The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
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So, if the two shorter sides are labelled as π and π, and the hypotenuse is labeled as π, we have the equation π squared plus π squared is equal to π squared.
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In our right triangle π΄ππΆ, the two shorter sides are ππΆ, which is 120, and π΄πΆ, which we donβt know, and the hypotenuse is 200.
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So, by applying the Pythagorean theorem, we have the equation 120 squared plus π΄πΆ squared is equal to 200 squared.
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We can evaluate both 120 squared and 200 squared and then subtract 14400 from each side of the equation, giving π΄πΆ squared equals 25600.
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To solve for π΄πΆ, we square root both sides of the equation, taking only the positive square root as π΄πΆ is a length.
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We find that π΄πΆ is equal to 160.
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Finally then, recall that we were asked to find the length of the line segment π΄π΅.
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And as π΄πΆ is half of π΄π΅, we can find π΄π΅ by doubling the value weβve just found.
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The length of π΄π΅ is, therefore, two multiplied by 160, which is 320.
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Including the appropriate units then, we have that the length of the line segment π΄π΅ is 320 centimeters.
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Itβs also worth pointing out that we could have worked in triangle π΅πΆπ rather than triangle π΄πΆπ.
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The Line ππ΅ is also a radius of the circle just like the line ππ΄.
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And so, these two triangles are congruent to one another.
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We could have applied the Pythagorean theorem in this triangle.
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And it would have given exactly the same result.