If three, two is the midpoint of the segment with endpoints at negative three, negative one and 𝑎, 𝑏, what is the value of 𝑎 plus 𝑏?
In order to calculate the midpoint of any two coordinates or points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, we can use the following equations. The 𝑥-coordinate can be calculated by adding 𝑥 one and 𝑥 two and dividing by two. The 𝑦-coordinate is equal to 𝑦 one plus 𝑦 two divided by two. In this question, the endpoints are negative three, negative one and 𝑎, 𝑏. And the midpoint is three, two.
Let’s firstly consider the 𝑥-coordinates. Substituting in our 𝑥-values gives us negative three plus 𝑎 divided by two is equal to three. We can multiply both sides of this equation by two. This gives us negative three plus 𝑎 is equal to six. Adding three to both sides of this equation gives us a value for 𝑎 equal to nine. The 𝑥-coordinate of the second point is nine.
Let’s now consider the 𝑦-coordinate. Substituting in these values gives us negative one plus 𝑏 divided by two is equal to two. Once again, we begin by multiplying both sides by two. This gives us negative one plus 𝑏 is equal to four. Our final step here is to add one to both sides of the equation. This gives us a value for 𝑏 equal to five. The 𝑦-coordinate of the second endpoint is five. The two endpoints that have a midpoint of three, two are negative three, negative one and nine, five.
We could also have found the second point by looking at the 𝑥𝑦 plane. We were told that one of the endpoints had coordinate negative three, negative one. We go along the 𝑥-axis to negative three and down the 𝑦-axis to negative one. We were also told that the midpoint had coordinate three, two. This is six units right and three units up from our endpoint negative three, negative one. As three, two is the midpoint of the segment, we need to move another six units right and three units up to find the other endpoint. This takes us to the coordinate nine, five. We have once again proved that the coordinate three, two is the midpoint of two endpoints, negative three, negative one and nine, five.
This isn’t the end of the question though as we were asked to work out the value of 𝑎 plus 𝑏. We know that 𝑎 was equal to nine. And 𝑏 was equal to five. This means that 𝑎 plus 𝑏 is equal to nine plus five. This gives us a final answer of 14.