The Perpendicular Bisector Theorem in Geometry
In geometry, the perpendicular bisector theorem states that if a line segment is bisected by a line that is perpendicular to the segment, then the two halves of the segment are equal in length. In other words, if you draw a line through the midpoint of a line segment that is perpendicular to the segment, then the two resulting pieces will be exactly the same length.
This theorem applies to any line segment, whether it's part of a larger shape or not. For example, you can use the perpendicular bisector theorem to find the length of one side of a rectangle if you know the lengths of the other three sides. You can also use it to split a triangle into two equal halves. Keep reading to learn how to use the perpendicular bisector theorem to solve for missing lengths and prove geometric relationships!
Solving for Missing Lengths with the Perpendicular Bisector Theorem
One common use for the perpendicular bisector theorem is solving for missing lengths in geometric figures. Suppose you're given a rectangle with sides that measure 3 cm and 5 cm, as shown in Figure 1 below.
Figure 1: A rectangle with sides 3 cm and 5 cm.
If you want to find the length of Side AC (the hypotenuse), you can draw a line through points B and C that is perpendicular to Side AC, as shown in Figure 2.
Figure 2: A line through points B and C that is perpendicular to Side AC.
By definition, this new line must pass through point D, which is the midpoint of Side AC. Since we know that Line DC is perpendicular to Side AC, we can use the perpendicular bisector theorem to solve for its length. We know that each half of Side AC (Line CD and Line AD) has a length of 4 cm. So, all we need to do is add 4 cm + 4 cm = 8 cm to get Side AC's length.
As another example, suppose you're given Triangle ABC shown in Figure 3 below.
Figure 3: Triangle ABC with ... drawn in.
To find Side BC's length, start by drawing a line through point A that is perpendicular to Side BC (see Figure 4).
Figure 4: A line through point A ... BC.
Then use what you know about right triangles—specifically, that the sum of a triangle's interior angles must equal 180 degrees—to solve for angle CBD using basic algebra (i.e., solve for x). You should find that . By substituting back into , we get . So Line AB has a length of 10 units and Line BC has a length of 6 units.
You can use similar methods to split any triangle into two equal halves! Just remember that one leg of each resulting triangle will always be , where b equals half of Line AB's length and h equals half of Line AC's length (see Figure 5).
Figure 5: The newly created ... have legs b and h.
Now let's move on to another common application for the perpendicular bisector theorem—proving geometric relationships!
Proving Geometric Relationships with the Perpendicular Bisector Theorem Below are some examples of commonly used geometric relationships that can be proven with this theorem:
- If two lines intersect at right angles, then they are perpendicular bisectors of each other's segments (see Figure 6).
- If two lines are parallel, then they cannot intersect (see Figure 7).
- If two lines intersect at right angles and are parallel, then they are both perpendicular bisectors of each other's segments (see Figure 8).
Figure 6: Lines l1 and l2 ... each other's segments.
Figure 7: Lines l3 and l4 ... because they're parallel!
Figure 8: Lines l5 and l6 ... each other's segments *and* they're parallel! As you can see from these examples, there are lots different ways that you can put the perpendicular bisector theorem to use in your geometry proofs! reinforcements-icon Created with Sketch. Key Takeaways To review, here are some key points from this lesson: - The perpendicular bisector theorem states that if a line segment is cut by a line perpendicularly, then each resulting piece will be equal in length. - This theorem applies whether or not the line segment is part of a larger shape or not! - You can use this theorem when solving for missing lengths or proving geometric relationships like parallelism or perpendicularly." Conclusion There you have it—a crash course on everything you need to know about using the perpendicular bisector theorem in geometry!"
FAQ
How do you find the perpendicular bisector theorem?
There are a few different ways that you can find the perpendicular bisector theorem. One way is to use the Pythagorean theorem. Another way is to use a straightedge and compass.
Why is the perpendicular bisector theorem true?
The perpendicular bisector theorem is true because it is a direct consequence of the Pythagorean theorem. If you have a right triangle, then the length of the hypotenuse is equal to the sum of the lengths of the other two sides. This means that the perpendicular bisector of the hypotenuse must pass through the midpoint of the other two sides.
What is the perpendicular bisector theorem and its converse?
The perpendicular bisector theorem states that the perpendicular bisector of a line segment passes through the midpoint of the line segment. The converse of the theorem states that if a line passes through the midpoint of a line segment, then it is the perpendicular bisector of the line segment.
What is a perpendicular bisector in simple terms?
A perpendicular bisector is a line that passes through the midpoint of a line segment and is perpendicular to the line segment.