Question

In this triangle, D is the midpoint of AB and E is the midpoint of BC. If DE = 20 = 1 and AC 22 +3, what is the length of AC ? A) 2.5 B) C) 4,5 ) D) 6 E) D A

Answer 1

The triangles BED and BCA are similar triangles because the sides BD and BA are on the same line, also BE and BC are on the same line.

Since the triangles are similar, there must be a proportion between their sides.

We are told by the instructions that D is the midpoint of AB, this means that AB is double the size of BD, and this proportion must remain in all of the sides from the small triangle to the bigger triangle.

That means that AC is double the size of DE:

`AC=2(DE)`

We know that AC=2x+3 and that DE=2x-1, thus, substituting those values in the previous equation:

`2x+3=2(2x-1)`

From this equation, we need to find x.

To solve for x:

-apply the distributive property on the right side of the equation:

`2x+3=4x-2`

-subtract 2x to both sides of the equation:

`3=4x-2x-2`

-combine the like terms on the right side:

`3=2x-2`

-add 2 to both sides of the equation:

`\begin{gathered} 3+2=2x \\ 5=2x \end{gathered}`

-divide both sides of the equation by 2:

`\begin{gathered} (5)/(2)=(2x)/(2) \\ 2.5=x \end{gathered}`

Now that we know the value of x, we can find the value of AC:

`\begin{gathered} AC=2x+3 \\ \text{substituting x=2.5} \\ AC=2(2.5)+3 \\ AC=5+3 \\ AC=8 \end{gathered}`

Answer: E)8