+471 Suppose C(-4,5) is the midpoint of Line AB and the coordinates of A are (2,17). Find the coordinates of B.
Thanks!
+2446 Hello, Mr.Owl. I hope you are doing well.
Generally, when we face problems that involve the midpoint, we are generally tasked with identifying the midpoint. This is not the case here. I have provided a nice visual of the midpoint formula in action.
Coordinate A with coordinates \((x_1,y_1)\) on the above diagram has \((2,17)\). Therefore, \(x_1=2\text{ and }y_1=17\). The midpoint, coordinate C, represented as \(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\) on the visual, has coordinates \((-4,5)\). This means that \(\frac{x_1+x_2}{2}=-4\text{ and }\frac{y_1+y_2}{2}=5\). Does this give you an idea on how to proceed?
+471 Is there a formula for finding half of the line?
x1= 2 x2= 17
y1=5 y2=5
A=P
C=M
Q=B
Q= (x2,y2)
From what you have said, I gather that point B is 5 and 17= (5,17)
Is that right or did I just drive off the cliff?
+2446 Be careful! I know that this is a lot of information to take in at once.
\(x_1=2\text{ and }x_2=?\\ y_1=17\text{ and }y_2=?\\ A=P\\ C=M\\ Q=B=(x_2,y_2)\\ \)
We know that \(\frac{x_1+x_2}{2}=-4\text{ and }\frac{y_1+y_2}{2}=5\). We just have to solve for the missing variables. That's all.
+471 Alright, so I have:
2+x2 divided by 2 = -4 so 2+ -10 divided by 2 = -4
17+ y2 divided by 2 = 5 so 17 + -7 divided by 2 = 5
So x2 = -10 and y2 = -7
So the answer should be (-10,-7)
+2446 Nice job, Mr.Owl.
You deserve a round of applause! Woohoo!
You asked about a formula regarding how to find the other endpoint when an endpoint and midpoint are given in the problem. I believe it exists, just for your knowledge.
If A is the endpoint at \((x_1,y_1)\) and C is at the midpoint at \((x_2,y_2)\), then B, the opposite endpoint, should be located at \((2x_2-x_1,2y_2-y_1)\). You will see that it works with the coordinates we have.
