Please help!!: What is the length of DE in the image provided? (Please explain so I can understand in the future)

A. DE=4
B. DE=5
C. DE=6
D. DE=7

Explanation:

We have these three facts:

Those three facts allow us to use the AAS (angle angle side) congruence theorem to prove that triangle ABC is congruent to triangle EBD.

Then by CPCTC (corresponding parts of congruent triangles are congruent), we know that AC = DE. Note how 'A' and 'C' are the first and last letters of ABC; similarly, 'E' and 'D' are the first and last letters of EBD. Segment ED is the same as segment DE.

So to find the length of DE, we need to find the length of AC. To do that, we need the value of x.

Recall earlier that AB = BE. Use this to find x

AB = BE

1.5x = 3x-6

1.5x-3x = -6

-1.5x = -6

x = -6/(-1.5)

x = 4

Now we can find the length of AC

AC = x+3

AC = 4+3

AC = 7

This must mean DE = 7 as well.

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