Lesson Video: Perpendicular Bisector Theorem and Its Converse Mathematics

In this video, we will learn how to use the perpendicular bisector theorem and its converse to find a missing angle or side in an isosceles triangle.

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Video Transcript

In this video, we will learn how to use the perpendicular bisector theorem and its converse to find a missing angle or side in a triangle. To do that, let’s think about the definitions we’re working with, starting with perpendicular. Two line segments, rays, lines, or any combination of those that meet at a 90-degree right angle are perpendicular. Perpendicular lines are lines that meet at a 90-degree angle. A bisector is an object — a line, ray, or line segment — that cuts another object, usually an angle or a line segment, into two equal parts. We can therefore say that a perpendicular bisector is a line, a ray or, a line segment that bisects another line segment at a right angle.

Now that we thought a bit about what a perpendicular bisector is, we can consider the perpendicular bisector theorem. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. Let’s see if we can sketch what this means. First, we’ll need a line segment, and then we have a perpendicular bisector. The perpendicular bisector forms a right angle with our line segment and divides the line segment in half.

Now we’re considering a point that is on this perpendicular bisector. This theorem is telling us that point will be equidistant from the endpoints of the line segment. On this diagram, that is showing that the lines created from our point to our endpoint, the yellow lines, will be equal to each other. One way to prove that this is true is to consider the two smaller triangles created by this perpendicular bisector. To do this, let’s go ahead and label the points on the triangle.

We have the larger triangle, triangle 𝐴𝐵𝐶. We then have the smaller triangle, triangle 𝐴𝐵𝐷, and the smaller triangle, triangle 𝐴𝐶𝐷. Because these two smaller triangles share their third side, we can say line segment 𝐴𝐷 is congruent with line segment 𝐴𝐷. We’ve shown they have a congruent side. And because we’re dealing with a perpendicular bisector, we know that line segment 𝐵𝐶 is congruent to line segment 𝐷𝐶, which means we have two congruent sides in these triangles. And between those two congruent sides, we have two congruent angles. Angle 𝐴𝐷𝐵 is congruent to angle 𝐴𝐷𝐶. They’re both 90-degree angles.

Using the side-angle-side theorem of triangles, we confirm that these two smaller triangles are congruent. And that’s another way of confirming that these two sides must be equal to each other. It’s a confirmation of the perpendicular bisector theorem. Here’s one place you might see something like this played out. If we say that our radio tower is the perpendicular bisector, we certainly want the radio tower to be forming a right angle with the ground. As you don’t want it leaning to the left or to the right, you’ll notice that it has what’s called guy wires, which add stability to the free standing structure.

If the point on the ground where we attach the guy wires are of equal distance from the perpendicular bisector, then we know that the wires on either side will be equal to each other in length. It does not mean that every set will be equal in length. Each set of wires will be equal in length to the other wire that goes to the same point on the tower.

Before we move on and look at some examples, we also want to consider the converse of the perpendicular bisector theorem. Remember, to find the converse means switching the hypothesis and conclusion of a conditional statement. In the case of the perpendicular bisector theorem, its converse says this. If a point is equidistant from the endpoints of a line segment, then the point is on the perpendicular bisector of the segment. If we start with our point and that point is equidistant to each endpoint of the line segment, this point must fall along the perpendicular bisector.

While the converse might not immediately seem true, one way to show this is to consider when the point is not equidistant from both endpoints. Imagine if this was our point and then we drew lines to the endpoints. We can see very clearly that these two line segments are not equal to one another. In addition to that, even if we drew a line that was a right angle with the base, it could not be a bisector because we know that a bisector must divide the line segment that it intersects in half. This confirms that in order for a point to fall on the perpendicular bisector of a segment, it must be equidistant from the endpoints of that line segment. Now we’re ready to consider some example problems.

In the following figure, find the length of line segment 𝑊𝑌.

The first thing we wanna do here is take stock of what we’re given in the figure. We have a triangle 𝑋𝑍𝑊. The length of line segment 𝑋𝑊 is equal to the length of line segment 𝑋𝑍, which is equal to 17. We could also say that this is therefore an isosceles triangle. In addition to that, we know that angle 𝑋𝑌𝑍 is a right angle. Based on this given information, we can draw some conclusions. Because we know that line segment 𝑋𝑊 is equal in length to line segment 𝑋𝑍 and we know that angle 𝑋𝑌𝑍 is 90 degrees, we can say that line segment 𝑋𝑌 is a perpendicular bisector.

We can make this claim based on the converse of the perpendicular bisector theorem. That tells us if a point is equidistant from the endpoints of a segment — for us, the point 𝑋 is equidistant from 𝑊 and 𝑍 — then the point is on the perpendicular bisector of the segment, which means we can say that line segment 𝑌𝑍 will be equal in length to line segment 𝑊𝑌, because of the definition of the perpendicular bisector, which divides the line segment it intersects in half. Because line segment 𝑌𝑍 equals 11, we can say that line segment 𝑊𝑌 will also be equal to 11.

We’re ready to look at another example.

In the diagram, 𝐴𝐵 equals six and 𝐵𝐷 equals five. Find 𝐴𝐶 and find 𝐶𝐷.

A good place to start is listing out the information we’re given. We have triangle 𝐴𝐵𝐶. And in that triangle, line segment 𝐵𝐷 is equal to line segment 𝐷𝐶. We’ve also been told that the measure of angle 𝐴𝐷𝐶 equals 90 degrees. It’s a right angle. We also know that 𝐴𝐵 measures six and 𝐵𝐷 measures five. From this information, we can draw some conclusions. First of all, we already know that line segment 𝐷𝐶 is equal in length to line segment 𝐵𝐷, which means we can say that line segment 𝐷𝐶 also measures five. And we can write the line segment either way. 𝐶𝐷 will be equal to 𝐷𝐶.

So we can say, first of all, that 𝐶𝐷 equals five. To find 𝐴𝐶, we’ll need to think a little bit more carefully about what we know. We know that the measure of angle 𝐴𝐷𝐶 is 90 degrees, and we know that the point 𝐷 is halfway between 𝐵 and 𝐶. This means we can say that 𝐴𝐷 is a perpendicular bisector of line segment 𝐵𝐶. We can say that this is true based on the definition of a perpendicular bisector. A perpendicular bisector has to divide the line segment in half and meet at a 90-degree angle.

And since we know that line 𝐴𝐷 is a perpendicular bisector of 𝐵𝐶, we can apply the perpendicular bisector theorem, which tells us that if a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints. Since 𝐴𝐷 is a perpendicular bisector, 𝐴𝐵 is equal in length to 𝐴𝐶. And if line segment 𝐴𝐶 is equal in length to line segment 𝐴𝐵, since line segment 𝐴𝐵 measures six, line segment 𝐴𝐶 will also measure six.

In our next example, we’ll use the properties of perpendicular bisectors to find a missing angle instead of a missing side length.

Find the measure of angle 𝐷𝐴𝐵.

When we’re given questions like this, it’s always good to start with what we’re given. We have triangle 𝐴𝐵𝐶. In this triangle, line segment 𝐴𝐶 is equal in length to line segment 𝐴𝐵. We know the measure of angle 𝐴𝐷𝐵 equals 90 degrees, and we know that the measure of angle 𝐶𝐴𝐷 equals 25 degrees. We want to know the measure of angle 𝐷𝐴𝐵. That’s this angle. To do that, we take the information we were given and draw some conclusions. Because line segment 𝐴𝐶 is equal to line segment 𝐴𝐵 and because the measure of angle 𝐴𝐷𝐵 is 90 degrees, we can say that line segment 𝐴𝐷 is a perpendicular bisector.

We based that on the converse of the perpendicular bisector theorem, which tells us that if a point is equidistant from the ends of two line segments — for us, that would be the line segments 𝐴𝐶 and 𝐴𝐵 that are equal — then the point 𝐴 must fall along the perpendicular bisector. Because line segment 𝐴𝐷 is a perpendicular bisector, we can say that line segment 𝐶𝐷 is equal in length to line segment 𝐵𝐷. We know that this perpendicular bisector creates two smaller triangles. And we can say that the smaller triangle 𝐴𝐷𝐶 must be congruent to the smaller triangle 𝐴𝐷𝐵.

We say this based on side-side-side congruence. Three sides of triangle 𝐴𝐷𝐶 are equal to the corresponding three sides of triangle 𝐴𝐷𝐵. And since these two triangles are congruent, we can say that the measure of angle 𝐶𝐴𝐷 will be equal to the measure of angle 𝐷𝐴𝐵. We’re saying these two angles are congruent, which makes the measure of angle 𝐷𝐴𝐵 25 degrees.

In our final example, we’ll need to use the properties of a perpendicular bisector to find the value of 𝑋.

In the diagram, line segment 𝐴𝐷 is the perpendicular bisector of line segment 𝐵𝐶. Find the value of 𝑥.

Since we know that line segment 𝐴𝐷 is a perpendicular bisector, we can add a few details to our diagram. The angle created at this intersection will be a right angle. And the line segment 𝐴𝐷 bisects line segment 𝐵𝐶, making 𝐵𝐷 equal to 𝐵𝐶. At this point, it does not seem like we have enough information to find the value of 𝑥, which is where the perpendicular bisector theorem comes into play. It says if a point is on the perpendicular bisector of a line segment — for us, that point could be 𝐴 — then that point is equidistant from the endpoints of the line segment. The endpoints of our line segment are 𝐵 and 𝐶. And 𝐴 is equidistant to point 𝐵 and point 𝐶, which makes line segment 𝐴𝐵 equal in length to line segment 𝐴𝐶.

And using this fact, we can set up an equation to solve for 𝑥. If 𝐴𝐵 is equal to 𝐴𝐶, then three 𝑥 plus one must be equal to five 𝑥 minus 12. And at this point, we need only to solve for 𝑥. First, we’ll subtract three 𝑥 from both sides to get both 𝑥-values on the same side of the equation. And we’ll have one equals two 𝑥 minus 12. From there, we add 12 to both sides of the equation. And we have the equation 13 equals two 𝑥, which means we need to divide both sides of the equation by two. 13 divided by two is six and a half, which means we can say that 𝑥 equals six and a half.

Before we finish, let’s do a quick review of the key points. We just have two main key points from this section. And that is the perpendicular bisector theorem tells us if a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. And then the converse of the perpendicular bisector theorem, which is also true, if a point is equidistant from the endpoints of a line segment, then the point is on the perpendicular bisector of the segment. These two properties often help us solve for missing side lengths or angle measures inside triangles.

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