Here' a quick summary of the answers:
Question a: center = (3, 2), scale = 9 / 20
Question b: center = (8, -1), scale = 3
Question c: center = (6, 4), scale = 3 / 2
Question d: center = (5, 3), scale = 2
Step-by-step explanation:
By "find the center of dilation", they're asking you to name the x/y point at which the image is scaled from.
The scale factor is the amount that the points are being scaled by to reach the "prime" of them (noted by the apostrophe). e.g., A would be the original point, and A' would be the transformed one.
To answer these, we'll need to pay careful attention to the coordinates in the graph. To get the scale, you can simply look at the equivalent side on both triangles, and compare the lengths.
In question A for example, we can see the line AC is 10 units wide, and the equivalent line A'C' is 4.5 units wide. This tells us that the triangle was scaled down by a factor of 4.5/10, or 9/20, or 45%
Finding out the center of dilation can be slightly more complicated. Note that there may be an easier way to do this that I am personally unaware of, but one way to do it would be to take the original and transformed versions of two points, e.g. A and A', and B and B', and use those points to define lines. We can then find where those lines intersect, and that tells us where the point of origin is.
Question A:
For example, in question A, the points A and A' are at (7, 4) and (5, 3) respectively. Remember that the slope of a line is equal to rise over run. In other words
s = (Ay - A'y) / (Ax - A'x)
s = (4 - 3) / (7 - 5)
s = 1/2, or 0.5
Now we can take that slope, and either of the the two points we used, and apply it to a point to define a line, using the formula Δy = sΔx
y - Ay = s(x - Ax)
y - 4 = 0.5(x - 7)
y = 0.5x - 3.5 + 4
y = 0.5x + 0.5
That gives us the line along which point A was translated when the triangle was scaled. Now let's find the same for point B:
s = (By - B'y) / (Bx - B'x)
s = (1 - 0) / (4 - 5)
s = 1 / -1
s = -1
And apply that slope to build our line:
y - By = s(x - Bx)
y - 0 = -1(x - 5)
y = -x + 5
Now that we have two lines created, we can simply equate them, solving for x, which tells us the x location of the center of dilation.
Given:
y = -x + 5
and
y = 0.5x + 0.5
we can say:
-x + 5 = 0.5x + 0.5
-2x + 10 = x + 1
-3x = -9
x = 3
So the center of dilation has an x coordinate of 3. To find it's y coordinate, we merely need to plug that into one of the line equations we already worked out:
y = -x + 5
y = -3 + 5
y = 2
So for question A the center of dilation is at (3, 2)
Question B:
This one is actually much easier, as we don't need to work out line equations to find the point of origin. The thing to notice here is that A and A' have the same x coordinate, forming a vertical line. Also B and B' have the same y coordinate, forming a horizontal line. All we need then is to know where those lines intersect. We can get that by taking the x coordinate of A and the y coordinate of B.
A lies at (8, 0), and B lies at (5, -1), so the center of dilation is (8, -1)
Using the same logic as above to find the scale, we get a scale of 3. This is easiest by comparing |CA| with |C'A'|, the left sides of the triangles. As you can see, that side is scaled up by a factor of 3.
Question C:
This one is far easier, as it is very obvious from the image that point A, at (6, 4) is the point of dilation. Using the same technique as above we can see that the scale is 3/2, or 1.5
Question D:
Again much easier, as we can see that the point of origin is on the halfway point of the side CA, (5, 3). Using the same method as above, we can see that the scale is 2.
