Original triangle a- 1,1 b- 2,3 c- 4,3

A) Reflection in y axis
B) Reflection in the line y= -1
C) Reflection y=2x

Now I think the first one would be A- -1,1 B- -2,3 C- -4,3.
The second I believe to be A- 1, -3 B-2, -5 C- 4, -5.
Can you please tell me if the first two are correct and what the third is. Thanks

2. On a coordinate plane, if a point with coordinates (a,b) is reflected about the y-axis, the reflected point's coordinates are (-a,b).

Similarly, on a coordinate plane, if a point with coordinates (a,b) is reflected about the x-axis, the reflected point's coordinates are (a,-b).

3. For the second problem, remember that the line of reflection is half-way between the image and the reflected reflected image.

More specifically, say you are given a point P=(a,b) on a coordinate plane. When P is reflected about the line y=k, whereby k is a constant, we will call the reflected point $\displaystyle P_r$. The x-coordinate will not change from P to $\displaystyle P_r$. What will change is the y-coordinate. If k>b, then the y-coordinate of $\displaystyle P_r$ will be $\displaystyle b+2(k-b)$. That is, the y-coordinate of $\displaystyle P_r$ will be the y-coordinate of P plus twice the distance between y-coordinate of P and k. In similar fashion, if k<b, then the y-coordinate will be shifting downward, i.e., the y-coordinate will be $$b-2(b-k) [tex]. So in your problem, you have a triangle that has points with coordinates A=(1,1), B=(2,3), and C=(4,3). Since -1 is smaller than each of the y-coordinates of the points of the triangle, you will be shifting the y-coordinates of A, B and C downward. I'll do the first. Since point A=(1,1) and -1<1, point \displaystyle A_r = (1, 1-2[1-[-1]]) = (1, -3) Note that I used \displaystyle A_r  simply to denote the point that is the reflection of point A. EDIT: Also, regarding the \displaystyle A_r = (1, 1-2[1-[-1]]) = (1, -3) bit, note that I used brackets in lieu of parenthesis in order to avoid the confusion that could occur when differentiating between coordinates-wrapping parenthesis and multiplication parenthesis. 4. For your third problem, again, I will get you started and show you how to compute the coordinates of the reflection of the first point, then you do the other two points. We have point A=(1,1), and we want to determine the coordinates of \displaystyle A_r , which is the point you get when you reflect point A over y=2x. We want to find the point that is on the opposite side of the line y=2x AND is equidistant along with point A to y=2x. How do we do this? First, let's compute the slope of the equation of the line perpendicular to \displaystyle y=2x . The slope of a perpendicular line takes half the negative reciprocal of the slope of the original line. Thus, the slope of the line perpendicular to \displaystyle y=2x  is \displaystyle y=-\frac{1}{2}.  Now, plug (1,1) into \displaystyle y=-\frac{1}{2}x + b . We get \displaystyle 1 = b - \frac{1}{2} \implies b=\frac{3}{2} . Next, we compute the intersection of the lines \displaystyle y=2x  and [mth] y=-\frac{1}{2}x + \frac{3}{2}$$, which is (1,2).

Since the x-coordinate of A is the same as the x-coordinate of (1,2), the x-coordinate of $\displaystyle A_r$ will not under go any change.

The y-coordinate of $\displaystyle A_r$ will jump up twice the difference between the y-coordinate of A and the y-coordinate of $\displaystyle A_r$; the y-coordinate of $\displaystyle A_r$ is $\displaystyle 1+2(2-1) = 3$.

So, $\displaystyle A_r = (1,3).$

-Andy

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# how to reflect a triangle in mathematics

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