Start by comparing the length of two corresponding sides.
Example Solution: Dilation with a scale factor of 12 followed by a reflection in y=x.
Practice makes perfect
To describe a similarity transformation that maps â–ł ABC to â–ł RST, let's begin by plotting the triangles with the given vertices.
Examining the two triangles, we can see that they differ in size. Therefore, we need to perform a dilation. To find the scale factor, we should compare two corresponding sides. Let's choose the shortest ones BC and ST. We can find their lengths by using the Distance Formula.
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
BC
( -2,0) ( -4,2)
sqrt(( -4-( -2))^2+( 2- 0)^2)
sqrt(8)
ST
( 0,-1) ( 1,-2)
sqrt(( 1- 0)^2+( - 2-( -1))^2)
sqrt(2)
Once we know the lengths, let's compare them. In our case the preimage is â–ł ABC and the image is â–ł RST.
k=ST/BC=sqrt(2)/sqrt(8)
Now, we can simplify the above ratio.
If we dilate â–ł ABC using a scale factor of 12, the triangles will have the same size. To perform this dilation, we should multiply the preimage's coordinates by 12.
(a,b)
(1/2a,1/2b)
A(6,4)
A'(3,2)
B(-2,0)
B'(-1,0)
C(-4,2)
C'(-2,1)
Let's plot the image â–ł A'B'C'.
We can see that the image â–ł A'B'C' does not have the same coordinates as â–ł RST. It appears as though there is a reflection across the line y=x. If this is true, the vertices will be such that the x - and y -coordinates are inverses.
(a,b)
(b,a)
A'(3,2)
(2,3)
B'(-1,0)
(0,-1)
C'(-2,1)
(1,-2)
It worked! We got the coordinates of △ RST. Indeed, y=x forms an axis of symmetry. Let’s reflect △ A'B'C' in the line.
We can conclude that the similarity transformation that maps â–ł ABC to â–ł RST is a dilation with a scale factor of 12 followed by a reflection in the line y=x.
