+743 Point $A$ is reflected over the line shown below to point $B$. Find the coordinates of $B$.
Write your answer as an ordered pair $(x,y)$.
The line is y = x + 3, and A = (7,1).
+135 Step one: Find the negative reciprocal of the line where it passes through A
The negative reciprocal of y = x+3 is y = -x+b, and to find b, we can plug in x and y. Doing so gets us "7 = -1 + b", so b is 8. Therefore, the line we were looking for is "y = -x+8"
Step two: Find the point of intersection of the lines.
To find the point of intersection, we simply find what value of x makes both equations true, or, in our case, what value of x makes "x + 3 = -x + 8". Solving for x, we get x = 2.5, so y = 5.5. Therefore, our lines intersect at (2.5, 5.5).
Step three: Find the point's distance from intersection, then flip that.
To go from (2.5, 5.5) to (7, 1), you must move right 4.5 units, then down 4.5 units. We then do the opposite of this, and move up 4.5 units and left 4.5 units from the intersection, bringing us to (2.5 - 4.5, 5.5+4.5) = (-2, 10), which is what we're looking for.
After following all these steps, we can now see that the coordinates of B are (-2, 10).
