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I have two points $ A = (3,5)$ and $B = (5,2)$, I want to find their midpoint C = (4, Y).

In this case Y would be 3.5

Is there a formula to find out what the $ Y $ of a point is, in relation to the other two points?

exemple: Given A (3.5) B (5.2) complete: C (4, Y), the answer would be: C (4.3.5); C (3.5, Y), the answer would be: C (3.5,4.25); C (4.5, Y), the answer would be: C (4.5,2.5);

ah Lives
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1 Answers1

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I think what you are trying to do is find the midpoint of the line joining $A$ and $B$. If we generalise and say that \begin{align}A&=(x_A,y_A)\\ B&=(x_B,y_B)\end{align}

then the coordinates of the midpoint of the line joining them together can be written as

$$C=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2}\right)$$

In the example you gave, we have \begin{align}A&=(3,5)\\ B&=(5,2)\end{align}

Therefore \begin{align}C&=\left(\frac{3+5}{2},\frac{5+2}{2}\right)\\ &=\left(\frac82,\frac 72\right)\\ &=(4,3.5)\end{align}

If in fact what you are trying to do is simply find the point on the line joining $A$ and $B$, given an $x$-coordinate, then we need to find the equation of the line.

We need to know the slope of the line which is given by $$m=\frac{y_B-y_A}{x_B-x_A}$$

Now we can use the "point slope formula" to find the equation of the line:

$$y-y_A=m(x-x_A)$$

Therefore, the $y$ coordinate can be found using

\begin{align}y&=m(x-x_a)+y_A\\ &=\frac{y_B-y_A}{x_B-x_A} (x-x_A)+y_A\\ &=\frac{(y_B-y_A)(x-x_A)}{x_B-x_A}+y_A\end{align}

For our example, this is the same as saying

\begin{align}Y&=\frac{(2-5)(x-3)}{5-3}+5\\ &=\frac{-3(x-3)}{2}+5\end{align}

When we have $C=(4,Y)$, then this gives

\begin{align}Y&=\frac{-3(4-3)}{2}+5\\ &=\frac{-3\times 1}{2}+5\\ &=5-\frac 32\\ &=3.5\end{align}

or, for $C=(3.5,Y)$, we would have

\begin{align}Y&=\frac{-3(3.5-3)}{2}+5\\ &=\frac{-3\times0.5}{2}+5\\ &=\frac{-1.5}2+5\\ &=5-0.75\\ &=4.25\end{align}

This is slightly different to the result you are expecting, but I believe your result is wrong and this is what you intended

lioness99a
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  • But what if X is 3.5? the Y would be 4.5, but which formula do I use? – ah Lives Mar 17 '20 at 15:41
  • Why would $X$ be $3.5$? Either you are trying to find the mid point or you are not, in which case you need to add some more details about what you actually want to know – lioness99a Mar 17 '20 at 15:43
  • I have two vertices, A and B, which form a line, I have an X number that can be any number between Ax and Bx, so I want to find the Y, ex: Given A (3.5) B (5.2) complete: C (4, Y), the answer would be: C (4.3.5) C (3.5, Y), the answer would be: C (3.5,4.5) C (4.5, Y), the answer would be: C (4.5,2.5) – ah Lives Mar 17 '20 at 15:51
  • With your A and B, having X=3.5 would give you Y=4.25 if you’re trying to find the corresponding point on the line joining A and B – lioness99a Mar 17 '20 at 17:45
  • I've updated my answer with another option which may answer your quesiton – lioness99a Mar 18 '20 at 08:29
  • Yes, thanks you, to this is what I needed, but I didn't know where to find the answer, you helped me a lot! ^ - ^ – ah Lives Mar 18 '20 at 14:23