Describe a similarity transformation that maps △ABC
to △RST

A(6, 4), B(−2, 0), C(−4, 2)and R(2, 3), S(0,−1), T(1,−2)

Fill in the blanks below. Please show all the work on how you got your answer.

Let's start with a picture.

We see RST is smaller, and BC is parallel to but in the opposite direction to its corresponding segment ST.  Both have slope -1.

If we look at the difference of points (technically called vectors but we don't have to go there) we get

C-B=(-2,2)

T-S=(1,-1)

Without further calculation we can see T-S is half the length of C-B.

The problem asks for a dilation followed by a reflection.   We know the dilation scale is k=1/2 because the new triangle is half the size.  

After dilation we get A'B'C':

A'(3,2), B'(-1,0), C'(-2,1)

We see now we need a reflection that flips the coordinates x and y.  That's the +45° line through the origin, namely y=x.

k=1/2, y=x

  see second attachment

Step-by-step explanation:

ΔRST is smaller, so the scale factor must be less than 1. The only available answer choice is k = 1/2.

When you scale ΔABC by 1/2, you get A'(3, 2), B'(-1, 0), C'(-2, 1). Comparing these coordinates to R, S, T, we find the x-, and y-values are swapped. This indicates a reflection across the line y=x.

____

In the graph, ΔA'B'C' is purple, ΔRST is blue.

i think 5 miles per hour

step-by-step explanation:

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240