# Calculating the Midpoint - Problem 4

It is also possible to use the midpoint formula to find one of the endpoints of the line segment, when given the other endpoint, (x_{1}, y_{1}). The goal is to find the other endpoint, (x_{2}, y_{2}). The midpoint formula states that you can find the midpoint (x, y) by finding the values of x = (x_{1} + x_{2})/2 and y = (y_{1} + y_{2})/2. Since you know the values of x_{1}, y_{1}, x, and y, it is possible to use algebra to solve for x_{2} and y_{2}. By manipulating the formula, you will find that x_{2} = 2x - x_{1}, and y_{2} = 2y - y_{1}. So, using this, you can plug in the known numbers to find the second endpoint.

Sometimes you might be asked to use the midpoint formula when you are doing any number something that causes geometry students to shake in their boots. Here we‘ve got an end point at AB we know the midpoint and we want to find the other endpoint in terms of the midpoint and the endpoint that you know.

So I’m not going to draw a picture here because we don’t where these pints are but I’m going to start by writing my formulas. The first one says the x coordinate of your midpoint is equal to x1 plus x2 divided by 2. So what are we solving for here? Well we're asking to find the endpoint x2 and y2 so I’m trying to solve for that variable x2.

I know x is X, that’s pretty easy and I know x1 is a so I’m going to substitute an a for x1. So we have a plus x2 divide by 2. Now all you're doing here is symbolic manipulation which means we are moving variables around. Now we're going to multiply by 2, 2 times x is just 2x we have a plus x2. So what happens to those 2’s there? 2 divided by 2 that’s a big curvy 1. Last to solve we are going to subtract a from both sides and I’m going to switch this around and say that x2 is equal to 2x minus a.

So looking at how we solve this is exactly how we are going to solved for y. Y is equal to y1 plus y2 all divided by 2 and again midpoint’s at y so that part is pretty easy we know our y1 is b so we are going to substitute in b for y1.

So b plus y2 divide by 2 and it looks like it’s basically the same process multiply by 2, multiply by 2, 2 times y is just 2y. Our 2's have equaled 1 there so 2y equals b plus y2 and the last step is to subtract b on both sides of the equation. These are not like terms so I’m going to say that y2 is equal to 2y minus b.

So we found our x2 and our y2 and I’m going to write this in an ordered pair because that’s what the problem is asking for. We said x2 is 2x minus a and our y2 is 2y minus b, and draw a box around it so your teacher can give you full credit.

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