Which coordinates could be the vertices of a triangle similar to triangle RST?

{original triangle coordinates R(0,0); S(0,4); T(3,0)}

a. (2,2), (2,6), and (6,0)***
b. (0,0), (0,6), and (10,0)
c. (2,2), (2,10), and (6,2)
d. (0,0), (0,8), and (6,0)

Please help Steve!

RST has sides RS=4, RT=3, ST=5

If the other triangles are ABC, then we have

(a) AB=4, AC=√20, BC=√20
As you can see, the various sides are not all the same multiple of RST's sides.

You will need to check the others to see whether their sides are in the ratio 4:3:5 as well.

To determine if a triangle is similar to triangle RST, you need to check if the ratios of the corresponding side lengths of the two triangles are equal. Let's calculate the ratios of the side lengths for each option:

a. (2,2), (2,6), and (6,0)
- Length of RS = √[(2-2)^2 + (6-2)^2] = 4
- Length of ST = √[(6-2)^2 + (0-6)^2] = 8.485
- Length of TR = √[(2-6)^2 + (2-0)^2] = 4.472

Ratio of side lengths RS:ST:TR = 4:8.485:4.472

b. (0,0), (0,6), and (10,0)
- Length of RS = √[(0-0)^2 + (6-0)^2] = 6
- Length of ST = √[(10-0)^2 + (0-6)^2] = 10
- Length of TR = √[(0-10)^2 + (0-0)^2] = 10

Ratio of side lengths RS:ST:TR = 6:10:10

c. (2,2), (2,10), and (6,2)
- Length of RS = √[(2-2)^2 + (10-2)^2] = 8
- Length of ST = √[(6-2)^2 + (2-10)^2] = 8.485
- Length of TR = √[(2-6)^2 + (2-2)^2] = 4

Ratio of side lengths RS:ST:TR = 8:8.485:4

d. (0,0), (0,8), and (6,0)
- Length of RS = √[(0-0)^2 + (8-0)^2] = 8
- Length of ST = √[(6-0)^2 + (0-8)^2] = 10
- Length of TR = √[(0-6)^2 + (0-0)^2] = 6

Ratio of side lengths RS:ST:TR = 8:10:6

By comparing the ratios, we can see that only option a has the same ratio of side lengths as triangle RST.

Therefore, the correct answer is a. (2,2), (2,6), and (6,0).