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LammettHash
Triangles ADE and ABC are similar, so their legs are proportional to one another. Let [tex]x[/tex] be the length of AE. You have

[tex]\dfrac{AD}{AE}=\dfrac{AB}{AC}\implies\dfrac6x=\dfrac7{x+3}[/tex]

Solve for [tex]x[/tex].

[tex]\dfrac6x=\dfrac7{x+3}[/tex]
[tex]6(x+3)=7x[/tex]
[tex]6x+18=7x[/tex]
[tex]x=18[/tex]

So the length of AE is 18.
akposevictor

Considering that triangle ADE and ABC in the diagram given are similar, the length of AE is: 18.

Recall:

  • Two triangles that are similar have corresponding sides that are congruent to each other as they have same shape.

Triangles ADE and ABC are similar.

  • Therefore:

AD/AB = AE/AC

AD = 6 in.

AB = 6 + 1 = 7

AE = ?

AC = AE + 3

  • Substitute

[tex]\frac{6}{7} = \frac{AE}{AE + 3}[/tex]

  • Cross multiply

[tex]6(AE + 3) = (7)(AE)\\\\6AE + 18 = 7AE\\\\[/tex]

  • Collect like terms

[tex]6AE + 18 = 7AE\\\\18 = 7AE - 6AE\\\\18 = AE\\\\\mathbf{AE = 18}[/tex]

Therefore, considering that triangle ADE and ABC in the diagram given are similar, the length of AE is: 18.

Learn more here:

https://brainly.com/question/10269775

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