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In this video, we will learn how to use the theory of the perpendicular bisector of a chord from the center of a circle and its converse to solve problems.
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Letβs begin by recalling how we can define a radius, a chord, and a diameter.
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A radius is a line segment which has one end at the center of the circle and the other on the circumference.
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We define a chord as any straight line segment whose endpoints both lie on the circumference of the same circle.
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The diameter is a special type of chord which passes through the center of the circle.
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We can also consider this to be made up of two radii.
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We will now consider what a perpendicular bisector of a chord looks like.
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In the diagram drawn, we have the chord π΅πΆ together with its perpendicular bisector.
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We will look at three theorems in this video.
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And in each case, we need to consider the center of the circle π΄ together with the radii π΄π΅ and π΄πΆ.
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Our first theorem states that if we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and bisects the chord π΅πΆ is perpendicular to π΅πΆ.
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The second theorem is very similar.
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If we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and is perpendicular to π΅πΆ also bisects π΅πΆ.
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Our third theorem is the converse to the chord bisector theorem.
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This states that if we have a circle with center π΄ containing a chord π΅πΆ, then the perpendicular bisector of π΅πΆ passes through π΄.
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It is important to note that the perpendicular bisector of a chord creates two congruent triangles.
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In the diagram drawn, these are triangles π΄π·πΆ and π΄π·π΅.
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We will now look at some examples where we can use the theorems discussed to find missing lengths and angles.
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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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In the diagram shown, we have a circle with center π.
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The chord π΄π΅ is bisected by the line segment ππ· at the point πΆ.
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Applying the chord bisector theorem, which states that if we have a circle with center π containing a chord π΄π΅, then the straight line that passes through π and bisects the chord π΄π΅ is perpendicular to π΄π΅.
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We can say that the measure of angle ππΆπ΅ is 90 degrees.
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We are told that the length of π΄π is 200 centimeters.
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And since this is a radius of the circle, π΅π is also 200 centimeters.
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We are also told that ππΆ is equal to 120 centimeters.
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Adding these measurements to our diagram, we have a right triangle ππΆπ΅ as shown.
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Applying the Pythagorean theorem, the length of π΅πΆ is equal to the square root of 200 squared minus 120 squared.
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This is equal to 160.
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As π΅πΆ is equal to 160 centimeters and the point πΆ bisects the line segment π΄π΅, then π΄πΆ is also equal to 160 centimeters.
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The line segment π΄π΅ is therefore equal to 320 centimeters.
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In our next example, we will see how we can apply the theroems to find the area of a triangle.
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In the figure below, if ππ΄ is equal to 17.2 centimeters and π΄π΅ is equal to 27.6 centimeters, find the length of the line segment ππΆ and the area of triangle π΄π·π΅ to the nearest tenth.
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Since π is the center of the circle and the line segment ππ· bisects the chord π΄π΅ at πΆ, we can apply the chord bisector theorem.
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This states that if we have a circle with center π containing a chord π΄π΅, then the straight line that passes through π and bisects π΄π΅ is perpendicular to π΄π΅.
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This means that the measure of angle ππΆπ΄ is 90 degrees.
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Since the chord π΄π΅ has length 27.6 centimeters and we know that πΆ bisects this chord, π΄πΆ is equal to 27.6 divided by two.
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This is equal to 13.8 centimeters.
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We are also told that the radius ππ΄ is equal to 17.2 centimeters.
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We can therefore use the Pythagorean theorem in the right triangle ππΆπ΄ such that ππΆ is equal to the square root of 17.2 squared minus 13.8 squared.
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This is equal to 10.266 and so on.
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And rounding to the nearest tenth, the line segment ππΆ is equal to 10.3 centimeters.
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The second part of our question asks us to calculate the area of triangle π΄π·π΅.
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Clearing some space, we recall that the area of any triangle is equal to the length of its base multiplied by the length of its perpendicular height divided by two.
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We know that the base of our triangle π΄π΅ is equal to 27.6 centimeters.
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The perpendicular height πΆπ· will be equal to the length of ππ· minus the length of ππΆ.
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ππ· is the radius of the circle, and we know this is equal to 17.2 centimeters.
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Whilst we could use 10.3 centimeters for ππΆ, it is better for accuracy to use the nonrounded version: ππΆ is equal to 10.266 and so on.
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Subtracting this from 17.2, we see that the length of πΆπ· is 6.933 and so on centimeters.
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We can now calculate the area of triangle π΄π·π΅ by multiplying this by 27.6 centimeters and then dividing by two.
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This is equal to 95.683 and so on.
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Once again, we need to round to the nearest tenth, giving us an answer of 95.7 square centimeters.
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In our next example, we will see how we can use perpendicular bisectors of chords to find a missing angle.
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Line segments π΄π΅ and π΄πΆ are two chords in the circle with center π in two opposite sides of its center, where the measure of angle π΅π΄πΆ is 33 degrees.
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If π· and πΈ are the midpoints of the line segments π΄π΅ and π΄πΆ, respectively, find the measure of angle π·ππΈ.
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We begin by noticing that ππΈ and ππ· both pass through the center of the circle and that they bisect the chords π΄πΆ and π΄π΅, respectively.
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We can therefore apply the chord bisector theorem, which states if we have a circle with center π containing a chord π΄π΅, then the straight line which passes through π and bisects π΄π΅ is perpendicular to π΄π΅.
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This means that, on our diagram, the measure of angle ππΈπ΄ and the measure of angle ππ·π΄ are both equal to 90 degrees.
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We notice that π΄π·ππΈ is a quadrilateral.
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And we know that the angles in a quadrilateral sum to 360 degrees.
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This means that the measure of angle π·ππΈ which we are trying to calculate is equal to 360 minus 90 minus 90 minus 33.
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This is equal to 147 degrees.
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In our final example, we will find the perimeter of a triangle using perpendicular bisectors of chords.
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In a circle of center π, π΄π΅ is equal to 35 centimeters, πΆπ΅ is equal to 25 centimeters, and π΄πΆ is equal to 40 centimeters.
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Given that line segment ππ· is perpendicular to line segment π΅πΆ and line segment ππΈ is perpendicular to line segment π΄πΆ, find the perimeter of triangle πΆπ·πΈ.
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We are given in the question the length of the three sides of the triangle πΆπ΅π΄.
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We know that π΄π΅ is 35 centimeters, πΆπ΅ is 25 centimeters, and π΄πΆ is 40 centimeters.
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We have been asked to calculate the perimeter of triangle πΆπ·πΈ.
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We will do this by firstly proving that triangles πΆπ΅π΄ and πΆπ·πΈ are similar using the chord bisector theorem.
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We notice from the diagram that the line segments ππΈ and ππ· both pass through π and meet the chords π΄πΆ and πΆπ΅ at right angles.
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The chord bisector theorem states that if we have a circle with center π containing a chord π΅πΆ, then the straight line that passes through π and is perpendicular to π΅πΆ also bisects π΅πΆ.
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In our diagram, this means that the length of π΄πΈ is equal to the length πΈπΆ and the length πΆπ· is equal to the length π·π΅.
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It is also clear from the diagram that π΄πΆ is equal to two multiplied by πΈπΆ and πΆπ΅ is equal to two multiplied by πΆπ·.
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As the two triangles πΆπ΅π΄ and πΆπ·πΈ also share the angle πΆ, we have two corresponding sides in proportion and the angle between the two sides is congruent.
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This proves that the two triangles are similar.
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And in fact triangle πΆπ΅π΄ is larger than triangle πΆπ·πΈ by a scale factor of two, as the lengths of the corresponding sides are twice as long.
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Side π΄πΆ is equal to two multiplied by side πΈπΆ, πΆπ΅ is equal to two πΆπ·, and π΄π΅ is equal to two multiplied by πΈπ·.
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We can calculate the perimeter of triangle πΆπ΅π΄ by adding 40, 35, and 25.
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This is equal to 100 centimeters.
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The perimeter of triangle πΆπ·πΈ will therefore be equal to half of this.
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This is equal to 50 centimeters.
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We have now seen a variety of examples of how perpendicular bisectors of chords can be used to find missing lengths, angle measures, and other unknowns in problems involving circles.
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We will now recap the key points from this video.
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The chord bisector theorem can be summarized in three ways.
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Firstly, if we have a circle with center π΄ containing a chord π΅πΆ, then the straight line that passes through π΄ and bisects the chord π΅πΆ is perpendicular to π΅πΆ.
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In the same way, the straight line that passes through π΄ and is perpendicular to π΅πΆ also bisects π΅πΆ.
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The converse of these two states that the perpendicular bisector of the chord π΅πΆ passes through the center π΄.
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As already stated, these theorems can be used to find missing lengths, angle measures, and other unknowns in problems involving circles.