Explain why the triangles are similar, then solve for x and find the length of BE and CD.

Question

Please help on this problem, I am stuck on finding x. The answer I got was 2.6, but I am not sure if that is entirely correct. Thank you in advance.

Answer 1

BE II DC

∠A is the common angle.

m∠AEB = m∠ADC → corresponding angles

m∠ABE = m∠ACD → corresponding angles

ΔDAC ∼ ΔEAB ← similar triangles by AAA

3/(3 + 4.5) = x/(x + 6)

7.5x = 3x + 18

4.5x = 18

x = 4

EB = x = 4

DC = x + 6 = 4 + 6 = 10

Answer 2

You know they are similar due to the fact that the internal angles are the same. you know that because the 2 lines marked in red are parallel. So given that they are similar you can set up a simple ratio problem or similar triangles problem which takes 2 of the sides and compares them to 2 of the same sides of the other triangle.

In this case I would set it up as.

Side 1 triangle 1 / side 1 triangle 2 = side 2 triangle 1 / side 2 triangle 2

7.5 / 3 = (x+6) / x

this goes to

2.5 = 1 + 6/x

goes to

1.5 = 6/x

goes to

1.5x = 6

goes to x = 6 /1.5 = 4

so x = 4

to check

put back in to the original equation..I will flip both side to make the math easier..

3/4 = 7.5 / 10

this is true so CHECKED.. Done

x=4