What is the image of (-1,3) by a half turn about (1,2)?

Question

a. (3,1)
b. (1,-3)
c. (-1,-2)
d. (3,-1)

back up your answer por favor

Answer 1

Is that REALLY how you spell "please" in Spanish? I must have not heard it properly. The answer is A.

See:

When we rotate a point of coordinates (a, b) a half turn about point (x, y), the image becomes (x - [a-x], y - [b-y]). In this case, that answer would be (1 - [-1 - 1], 2 - [3 - 2]), or (3,1).

Additional details:

Take another example:

If we were to rotate point (4,1) about (5,8), this would give us (5 - [4 - 5], 8 - [1 - 8]), or (6, 15).

Check it. It's true!

NOTE that this is only verification, not proof. If you want proof, go to that other guy.

Answer 2

3, 1

Imagine a line that connects the points (-1,3) and (1,2).

1,2 is the pivot. That means that the new orientation will be along a line that includes that pivot point. The half turn is essentially movint the line 180degrees. So when you graph the line connecting thetwo points, the third point in the half turn will be flipped. The new point will move across 2 units (because you move two units from -1 to 1 andit will move down (since it is flipped) by one unit (the difference between 3 and 2). Add this to the pivot: 1+2=3, 2-1=1 and you get 3, 1.

Answer 3

Suppose the point is (x, y).
Then (1,2) is just the mid-point of the line joining (-1,3) and (x,y).
Therefore
1 = (-1+x)/2 and 2 = (3+y)/2
Solving these
2 = -1 + x or x = 3
4 = 3 + y or y = 1.
Hence the point is (3, 1)

Answer 4

a.
if you turn it...the x axis and y axis will switch.