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2.5 P) Midpoint Between Two Points
2.5 P) Midpoint Between Two Points
Sometimes it may be the case that you need to work out the midpoint between two points. We can work out the midpoint between two points by labelling the points as point 1 and point 2; it does not matter which point we label as point 1 and which point we label as point 2. We then sub the values into the formula below:
What this formula is doing is working out an average/ the mean for the x and y coordinates. When we are using this formula, we need to be really careful with the signs of the coordinates.
Let’s have a few examples.
Let’s have a few examples.
Example 1
Point A has the coordinates (3, 5) and point B has the coordinates (7, 11). What are the coordinates for the midpoint of A and B?
The first step in finding the coordinates of the midpoint is to label the coordinates up. It does not matter which of the points is 1 and which of the points is 2. I am going to have A as point 1 and B as point 2. The labelled coordinates are shown below.
Point A has the coordinates (3, 5) and point B has the coordinates (7, 11). What are the coordinates for the midpoint of A and B?
The first step in finding the coordinates of the midpoint is to label the coordinates up. It does not matter which of the points is 1 and which of the points is 2. I am going to have A as point 1 and B as point 2. The labelled coordinates are shown below.
We now sub the values for x1, y1, x2 and y2 into their places in the formula for working out the midpoint.
We now complete the calculations to find the coordinates of the midpoint of A and B.
This tells us that the coordinates for the midpoint of A and B are (5, 8).
Example 2
Point C has the coordinates (-3, 6) and point D has the coordinates (-5, -9). Work out the coordinates for the midpoint of C and D?
We are going to work out the midpoint of C and D in exactly the same way. The first step is to label the coordinates. I am going to have C as point 1 and D as point 2. The labelled up coordinates are
Point C has the coordinates (-3, 6) and point D has the coordinates (-5, -9). Work out the coordinates for the midpoint of C and D?
We are going to work out the midpoint of C and D in exactly the same way. The first step is to label the coordinates. I am going to have C as point 1 and D as point 2. The labelled up coordinates are
We now sub the values for x1, y1, x2 and y2 into their correct places in the formula for working out the midpoint. There are a few negative values to sub in and in order to ensure that I do not make any mistakes with the signs, I am going to place all of the negative values that are being added inside a bracket when they go into the formula.
The adding of a negative results in the number being taken. This means that -3 + (-5) becomes -3 – 5, and 6 + (-9) becomes 6 – 9.
We now finish off the working in the usual way.
This tells us that the midpoint is (-4, -1.5).
Whenever we have a question that asks us to find the midpoint, it is worth double checking the answer to see whether it looks right. We can check the answer by seeing whether the x and y coordinates for the midpoint that we have found are in between the x and y coordinates of the two points (C & D) that we are finding the midpoint of. The x coordinate that we found for our midpoint was -4, and -4 is between -3 (the x coordinate for C) and -5 (the x coordinate for D), thus implying that we have the correct x value for the midpoint. The y coordinate that we found for our midpoint was -1.5, and -1.5 is between 6 (the y coordinate for C) and -9 (the y coordinate for D), thus implying that we have the correct y value for the midpoint.
I would definitely recommend this checking process because it is a quick way to ensure that you have the correct answer. Also, it is especially worth doing when we have negative values for the coordinates because there is a greater chance of making a mistake when negative coordinates are involved. Finally, another way that you could check whether you have obtained the correct midpoint is to draw a quick sketch of a graph and plot the points (the midpoint and the two points that you are finding the midpoint of). If your midpoint looks like it is in between your two points that you are finding the midpoint of, it strongly suggests that you have found the correct coordinates for the midpoint.
Whenever we have a question that asks us to find the midpoint, it is worth double checking the answer to see whether it looks right. We can check the answer by seeing whether the x and y coordinates for the midpoint that we have found are in between the x and y coordinates of the two points (C & D) that we are finding the midpoint of. The x coordinate that we found for our midpoint was -4, and -4 is between -3 (the x coordinate for C) and -5 (the x coordinate for D), thus implying that we have the correct x value for the midpoint. The y coordinate that we found for our midpoint was -1.5, and -1.5 is between 6 (the y coordinate for C) and -9 (the y coordinate for D), thus implying that we have the correct y value for the midpoint.
I would definitely recommend this checking process because it is a quick way to ensure that you have the correct answer. Also, it is especially worth doing when we have negative values for the coordinates because there is a greater chance of making a mistake when negative coordinates are involved. Finally, another way that you could check whether you have obtained the correct midpoint is to draw a quick sketch of a graph and plot the points (the midpoint and the two points that you are finding the midpoint of). If your midpoint looks like it is in between your two points that you are finding the midpoint of, it strongly suggests that you have found the correct coordinates for the midpoint.