Find the Sides of Triangle DEF: A Challenge!

[CharlesLin]

I found this question in my study guide

triangle ABC is similar to triangle DEF. Triangle ABC has sides 4,6,8. Wich could be the corresponding sides of a triangle DEF?
Indicate all that apply

A) 1, 1.5, 2
B)1.5, 2.25, 3
C)6, 9, 12
D) 8, 12, 16
E)10, 15, 20

What I did was add the sides of ABC 4, 6, 8=18. I know then, that the addition of the sides of triangle DEF should be a factor of 18.

Following this line of thinking, my answer would be D) and E). Unfortunately my answer is incomplet, acording to my guide...
How can we approach this? What I'm not seeing?

 

[MarkFL]

If one shape if similar to another, then all corresponding sides will have the same scaling factor. For example, if we take the sides of $\triangle ABC$ and divide them all by 4, we get choice A).

Can you find the others now?

 

[CharlesLin]

thanks I think I understand. however I wounder if thers a way to calculate the scaling factor?

because how do you know which number to divide?

In other words how do I find the number that divides 4,6,8 and gives 1,1.5,2

 

[MarkFL]

thanks I think I understand. however I wounder if thers a way to calculate the scaling factor?

because how do you know which number to divide?

In other words how do I find the number that divides 4,6,8 and gives 1,1.5,2

I would look for a potential scaling factor $k$ by taking the first datum given for each triangle (the given triangle with which we are to compare the others and then each choice of triangles in turn) and divide the choice by the given. So, for example let's look at choice A):

\(\displaystyle k=\frac{1}{4}=0.25\)

And then we find:

\(\displaystyle 6k=1.5\)

\(\displaystyle 8k=2\)

And these match the other two sides of choice A), so we know A) is a similar triangle.

So, next let's look at choice B):

1.5, 2.25, 3

\(\displaystyle k=\frac{1.5}{4}=\frac{3}{8}=0.375\)

Then we find:

\(\displaystyle 6k=2.25\)

\(\displaystyle 8k=3\)

And so we know that choice B) is also similar. Can you do the comparisons for the remaining choices?

 

[CharlesLin]

so then we have

$\frac{6}{4}$=1.5=K

6(1.5)=9

8(1.5)=12 $\therefore$ C is a similar to triangle ABC$\frac{8}{4}$=4=K

2*6=12
2*8=16 D) is similar to ABC

$\frac{10}{4}$=2.5=K

2.5*6=15
2.5*8=20 E) is similar to ABC

thank you very much for helping with this one.