Midpoints and Congruent Segments - Problem 1


A midpoint divides a line segment into two congruent pieces, meaning that the lengths of those pieces are equal. So, if given expressions for these lengths, it is possible to find the length of a piece and the length of the entire line segment.

First, solve for the unknown variable by setting the two expressions equal to one another and solving using algebra. For example, if one piece is 3y - 1 and the other is 6y - 4, set 3y - 1 = 6y - 4, and solve for y. After solving for the variable, plug the value back in to one of the expressions to find the length of the piece. In this example, y=1, so plug in 1 for one of the expressions, 3y - 1. 3(1) - 1 = 2, so the length of the piece is 2.

There are a couple ways to find the length of the line segment. You could add the expressions for the two pieces together and plug in the value of the variable (for example, plug in y=1 into the expression 3y - 1 + 6y - 4 = 9y - 5 to find 9(1) - 5 = 4). Or, since we know that a midpoint divides the line segment into two equal parts, you can simply double the value of the length of the piece solved for earlier (we found that the length of one piece is 2 units, so the length of the entire line segment is 4).


Applying the midpoint theorem that a midpoint will divide a line segment into two congruent pieces, let’s look at a problem. If I tell you that z is the midpoint of this line segment x, y and I tell you that xz is 3y minus 1 and zy is 6y minus 4, I want to find the length of xz and the length of xy. But how do I start.

Well the first thing I know is that if this is a midpoint, then those two lines segments must be congruent. If we go back to what we just said, we can set those two equal to each other. So xz is 3y minus 1, so I’m going to write 3y minus 1 is equal to the other side 6y minus 4. And the reason why I can say this is because in the problem it says that z is the midpoint. Always use as much information as the problem can tell you. So now it just becomes an algebra problem.

I got variables on both sides so I need to eliminate one of those variables. I’m going to subtract 3y from both sides and I did that to keep my variable positive. So I have a -1 equals 3y minus 4 and again that negative belongs to that 1 so that’s why I kept it with it. To isolate the y term I need to add 4 to both sides. -1 plus 4 is +3, 3 equals 3y and to solve this we need to divide both sides by 3 which tells me that y must be 1.

Okay and we’re done right? No, we have to look at the problem. Xz and xy have not been found. We just know what the variable y is. So let’s start with xz.

Line segment xz is equal to 3y minus 1. So if I substitute in 1 for y then I can find out what xz is. So making that substitution I’m going to say 3 times 1 minus 1 is equal to xz. I want to make sure that you can see that minus sign. So 3 times 1 is 3, minus 1 is 2. Okay now usually your geometry teacher will give you units, since I didn’t give you units here we can just write xz line segment is equal to 2 units. It might be centimeters, it might be inches, something like that on your homework. We’re not done because we still need to know what xy is. Xy is the sum of these two segments.

So in my head what I’m going to do is I’m going to say xy is equal to 3y plus 6y which is 9y, all I’m doing is combining like terms, and -1 and -4 is -5. Okay so I combine like terms, I’m just adding up those two segments. I know y is 1 so substituting xy is equal to 9 times 1 minus 5 and 9 minus 5 is 4. And again we don’t know what our units are so I’m just going to write 4 units. So you have two answers here xy is equal to 4 units and xz is equal to 2 units.

congruent segments bisect