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Lesson Explainer: Midpoints and Perpendicular Bisectors Mathematics

In this explainer, we will learn how to find the perpendicular bisector of a line segment by identifying its midpoint and finding the perpendicular line passing through that point.

Our first objective is to learn how to calculate the coordinates of the midpoint of a line segment connecting two points. Suppose we are given two points 𝑃(π‘₯,𝑦) and 𝑃(π‘₯,𝑦). The midpoint of the line segment π‘ƒπ‘ƒοŠ§οŠ¨ is the point 𝑀 lying on π‘ƒπ‘ƒοŠ§οŠ¨ exactly halfway between π‘ƒοŠ§ and π‘ƒοŠ¨. This means that the π‘₯-coordinate π‘₯ of 𝑀 lies halfway between π‘₯ and π‘₯ and may therefore be calculated by averaging the two points, giving us π‘₯=π‘₯+π‘₯2ο‰οŠ§οŠ¨. The same holds true for the 𝑦-coordinate 𝑦 of 𝑀. This leads us to the following formula.

Formula: The Coordinates of a Midpoint

Suppose 𝑃(π‘₯,𝑦) and 𝑃(π‘₯,𝑦) are points joined by a line segment π‘ƒπ‘ƒοŠ§οŠ¨. Then, the coordinates of the midpoint of the line segment π‘ƒπ‘ƒοŠ§οŠ¨ are given by 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.ο‰ο‰οŠ§οŠ¨οŠ§οŠ¨

Let us practice finding the coordinates of midpoints.

Example 1: Finding the Midpoint of a Line Segment given the Endpoints

Given 𝐴(4,8) and 𝐡(6,6), what are the coordinates of the midpoint of 𝐴𝐡?

Answer

We recall that the midpoint 𝑀 of a line segment is the point halfway between the endpoints, which we can find by averaging the π‘₯- and 𝑦-coordinates of 𝐴 and 𝐡 respectively. Thus, we apply the formula: 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2=ο€Ό4+62,8+62=(5,7).ο‰ο‰οŠ§οŠ¨οŠ§οŠ¨

Therefore, the coordinates of the midpoint of 𝐴𝐡 are (5,7).

We can also use the formula for the coordinates of a midpoint to calculate one of the endpoints of a line segment given its other endpoint and the coordinates of the midpoint.

Example 2: Finding an Endpoint of a Line Segment given the Midpoint and the Other Endpoint

The origin is the midpoint of the straight segment 𝐴𝐡. Find the coordinates of point 𝐡 if the coordinates of point 𝐴 are (βˆ’6,4).

Answer

Here, we have been given one endpoint of a line segment and the midpoint 𝑀(0,0) and have been asked to find the other endpoint. We can do this by using the midpoint formula in reverse: 𝑀(π‘₯,𝑦)=(0,0)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2=ο€½βˆ’6+π‘₯2,4+𝑦2.ο‰ο‰οŒΊοŒ»οŒΊοŒ»οŒ»οŒ»

This gives us two equations: βˆ’6+π‘₯2=0βˆ’6+π‘₯=0π‘₯=6 and 4+𝑦2=04+𝑦=0𝑦=βˆ’4.

We conclude that the coordinates of 𝐡 are (6,βˆ’4).

One application of calculating the midpoints of line segments is calculating the coordinates of centers of circles given their diameters for the simple reason that the center of a circle is the midpoint of any of its diameters.

In the next example, we will see an example of finding the center of a circle with this method.

Example 3: Finding the Center of a Circle given the Endpoints of a Diameter

Points 𝐴(4,1) and 𝐡(βˆ’4,βˆ’5) define the diameter 𝐴𝐡 of a circle with center 𝑀. Find the coordinates of 𝑀 and the circumference of the circle, rounding your answer to the nearest tenth.

Answer

The center 𝑀 of the circle is the midpoint of its diameter 𝐴𝐡. Recall that the midpoint 𝑀 of a line segment (such as a diameter) can be found by averaging the π‘₯- and 𝑦-coordinates of the endpoints 𝐴 and 𝐡 as follows: 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2=ο€Ό4βˆ’42,1βˆ’52=(0,βˆ’2).ο‰ο‰οŒΊοŒ»οŒΊοŒ»

The circumference of a circle is given by the formula 𝐢=2πœ‹π‘Ÿ, where π‘Ÿ is the length of its radius. The length of the radius is the distance from the center of the circle to any point on its radius, for example, the point 𝐴(4,1). We can calculate this length using the formula for the distance between two points 𝑀 and 𝐴: π‘Ÿ=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)π‘Ÿ=(0βˆ’4)+(βˆ’2βˆ’1)=16+9=25.οŠ¨ο‰οŒΊοŠ¨ο‰οŒΊοŠ¨οŠ¨οŠ¨οŠ¨

Taking the square roots, we find that π‘Ÿ=5 and therefore the circumference is 2πœ‹π‘Ÿ=10πœ‹=31.4 to the nearest tenth.

In conclusion, the coordinates of the center are (0,βˆ’2) and the circumference is 31.4 to the nearest tenth.

We turn now to the second major topic of this explainer, calculating the equation of the perpendicular bisector of a given line segment.

Definition: Perpendicular Bisectors

Given a line segment 𝐴𝐡, the perpendicular bisector of 𝐴𝐡 is the unique line perpendicular to 𝐴𝐡 passing through the midpoint of 𝐴𝐡.

Recall that for any line 𝐿 with slope π‘š, the slope of any line perpendicular to it is the negative reciprocal of π‘š, that is, βˆ’1π‘š. We can use this fact and our understanding of the midpoints of line segments to write down the equation of the perpendicular bisector of any line segment.

How to: Calculating the Equation of the Perpendicular Bisector of a Line Segment

Suppose we are given a line segment π‘ƒπ‘ƒοŠ§οŠ¨ with endpoints 𝑃(π‘₯,𝑦) and 𝑃(π‘₯,𝑦) and want to find the equation of its perpendicular bisector.

  1. First, we calculate the slope of the line segment. To do this, we recall the definition of the slope: slopeofchangeinchangein𝑃𝑃=𝑦π‘₯=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯.
  2. Next, we calculate the slope of the perpendicular bisector as the negative reciprocal of the slope of the line segment: slopeofperpendicularbisector=βˆ’π‘₯βˆ’π‘₯π‘¦βˆ’π‘¦.
  3. Next, we find the coordinates of the midpoint of π‘ƒπ‘ƒοŠ§οŠ¨ by applying the formula to the endpoints: 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.ο‰ο‰οŠ§οŠ¨οŠ§οŠ¨
  4. We can now substitute these coordinates and the slope into the point–slope form of the equation of a straight line: (π‘¦βˆ’π‘¦)=π‘š(π‘₯βˆ’π‘₯)ο€Όπ‘¦βˆ’π‘¦+𝑦2=βˆ’π‘₯βˆ’π‘₯π‘¦βˆ’π‘¦ο€Όπ‘₯βˆ’π‘₯+π‘₯2.ο‰ο‰οŠ§οŠ¨οŠ¨οŠ§οŠ¨οŠ§οŠ§οŠ¨
    This gives us an equation for the perpendicular bisector.

Let us have a go at applying this algorithm.

Example 4: Finding the Perpendicular Bisector of a Line Segment Joining Two Points

Find the equation of the perpendicular bisector of the line segment joining points 𝐴(1,3) and 𝐡(7,11). Give your answer in the form 𝑦=π‘šπ‘₯+𝑐.

Answer

To find the equation of the perpendicular bisector, we will first need to find its slope, which is the negative reciprocal of the slope of the line segment joining 𝐴 and 𝐡. This is given by slopeof𝐴𝐡=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯=11βˆ’37βˆ’1=86=43.

Now, we can find the negative reciprocal by flipping over the fraction and taking the negative; this gives us the following: slopeofperpendicularbisector=βˆ’34.

Next, we need the coordinates of a point on the perpendicular bisector. Since the perpendicular bisector (by definition) passes through the midpoint of the line segment 𝐴𝐡, we can use the formula for the coordinates of the midpoint: 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2=ο€Ό1+72,3+112=(4,7).ο‰ο‰οŠ§οŠ¨οŠ§οŠ¨

Substituting these coordinates and our slope βˆ’34 into the point–slope form of the equation of a straight line, π‘¦βˆ’π‘¦=π‘š(π‘₯βˆ’π‘₯)π‘¦βˆ’7=βˆ’34(π‘₯βˆ’4), and rearranging into the form 𝑦=π‘šπ‘₯+𝑐, we have 𝑦=βˆ’34π‘₯+10.

For our last example, we will use our understanding of midpoints and perpendicular bisectors to calculate some unknown values.

Example 5: Determining the Unknown Variables That Describe a Perpendicular Bisector of a Line Segment

A line segment 𝐴𝐡 joins the points 𝐴(βˆ’6,βˆ’6) and 𝐡(0,𝑝). The perpendicular bisector of 𝐴𝐡 has equation 𝑦=βˆ’3π‘₯+𝑐. Find the values of 𝑝 and 𝑐.

Answer

Since the perpendicular bisector has slope βˆ’3, we know that the line segment 𝐴𝐡 has slope 13 (the negative reciprocal of βˆ’3). We can calculate the 𝑦-coordinate of point 𝐡 (that is, 𝑝) by using the definition of the slope π‘š: π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯13=π‘βˆ’(βˆ’6)0βˆ’(βˆ’6)=𝑝+662=𝑝+6𝑝=βˆ’4.

We will calculate the value of 𝑐 in the equation 𝑦=βˆ’3π‘₯+𝑐 of the perpendicular bisector using the coordinates of the midpoint 𝑀 of 𝐴𝐡 (which is a point that lies on the perpendicular bisector by definition).

We have the formula 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2=ο€Όβˆ’6+02,βˆ’6βˆ’42=(βˆ’3,βˆ’5).ο‰ο‰οŠ§οŠ¨οŠ§οŠ¨

We can now substitute π‘₯=βˆ’3 and 𝑦=βˆ’5 into the equation of the perpendicular bisector and rearrange to find 𝑐: 𝑦=βˆ’3π‘₯+π‘βˆ’5=βˆ’3Γ—βˆ’3+𝑐𝑐=βˆ’5βˆ’9=βˆ’14.

Our solution to the example is 𝑝=βˆ’4, 𝑐=βˆ’14.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • We can use the formula 𝑀(π‘₯,𝑦)=ο€Όπ‘₯+π‘₯2,𝑦+𝑦2οˆο‰ο‰οŠ§οŠ¨οŠ§οŠ¨ to find the coordinates of the midpoint of a line segment given the coordinates of its endpoints. We can use the same formula to calculate coordinates of an endpoint given the midpoint and the other endpoint.
  • We can calculate the centers of circles given the endpoints of their diameters.
  • We know that the perpendicular bisector of a line segment is the unique line perpendicular to the segment passing through its midpoint.
  • We have a procedure for calculating the equation of the perpendicular bisector of a line segment given the coordinates of its endpoints:
    1. We first calculate its slope as the negative reciprocal of the slope of the line segment.
    2. We then find the coordinates of the midpoint of the line segment, which lies on the bisector by definition.
    3. Finally, we substitute these coordinates and the slope into the point–slope form of the equation of a straight line, which gives us an equation for the perpendicular bisector.

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