Question

What is the length of AC?

Answer 1

we know that

If AC is parallel to MN

then

Triangle ABC and triangle MBN are similar and the ratio of the corresponding sides are equal

so

$\frac{AB}{MB}=\frac{AC}{MN}$

we have

$AB=8\ ft\\MB=4\ ft\\MN=9\ ft$

substitute

$\frac{8}{4}=\frac{AC}{9}$

$AC=2*9=18\ ft$

therefore

the answer is the option

$18\ ft$

Answer 2

The length of AC is $\fbox{\begin\\\ \bf 18 \text{ft}\\\end{minispace}}$, that is the $\fbox{\begin\\\ \bf option (4)\\\end{minispace}}$ is correct.

Further Explanation:

Similar triangles are of the same shape such that the ratios of corresponding sides are equal.

In similar triangles, size is not same and it can vary.

The given triangle is redrawn and attached below as Figure 1.

The $\triangle \text{ABC}$ and $\triangle \text{MBN}$ are similar to each other as shown in given Figure.

Therefore, the ratio of corresponding sides of each triangle is equal.

That is, $\fbox{\begin\\\ \dfrac{\text{AB}}{\text{MB}}=\dfrac{\text{CB}}{\text{NB}}=\dfrac{\text{AC}}{\text{MN}}\\\end{minispace}}$

Also, it is given that $\text{AM}=\text{MB}$ and $\text{BN}=\text{NC}$.

The length of AB is calculated as follows:

$\begin{aligned}\text{AB}&=\text{AM}+\text{MB}\\&=4+4\\&=8\end{aligned}$

Therefore, the length of side AB is $8\text{feet}$.

Similarly, the length of BC is calculated as follows:

$\begin{aligned}\text{BC}&=\text{BN}+\text{NC}\\&=3+3\\&=6\end{aligned}$

Therefore, the length of side BC is $6\text{feet}$.

Since the corresponding sides are proportional then,

$\begin{aligned}\dfrac{\text{AB}}{\text{MB}}&=\dfrac{\text{AC}}{\text{MN}}\\ \dfrac{8}{4}&=\dfrac{\text{AC}}{9}\\2&=\dfrac{\text{AC}}{9}\end{aligned}$

This can be further be solved to obtain the value of AC as,

$\begin{aligned}\text{AC}&=2\times9\\ \text{AC}&=18\end{aligned}$  

THerefore, the length of side AC is $18\text{feet}$.

Option (1)

In option (1) it is given that the length of the side AC is $3\text{feet}$.

As per our calculation the length of the side AC is $18\text{feet}$.

This implies that the option (1) is incorrect.

Option (2)

In option (2) it is given that the length of the side AC is $4\text{feet}$.

As per our calculation the length of the side AC is $18\text{feet}$.

This implies that the option (2) is incorrect.

Option (3)

In option (3) it is given that the length of the side AC is $9\text{feet}$.

As per our calculation the length of the side AC is $18\text{feet}$.

This implies that the option (3) is incorrect.

Option (4)

In option (4) it is given that the length of the side AC is $18\text{feet}$.

As per our calculation the length of the side AC is $18\text{feet}$.

This implies that the option (4) is correct.

Therefore, the length of the side AC is $\fbox{\begin\\\ \bf 18\text{feet}\\\end{aligned}}$ i.e., option (4) is correct.

Learn more:

1. A problem on triangle https://brainly.com/question/7437053

2. A problem on transformation of triangle https://brainly.com/question/2992432

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Triangles

Keywords: Triangles, angles, similar triangles, adjacent lines, parallel lines, proportional, lines,  Congruent triangle, sides.