# Geometry help?

Mar 30, 2018

$x = 16 \frac{2}{3}$

#### Explanation:

$\triangle M O P$ is similar to $\triangle M L N$ because all the angles of both triangles are equal.
This means that the ratio of two sides in one triangle will be same as that of another triangle so $\text{MO"/"MP"="ML"/"MN}$
After putting in values, we get x/15=(x+20)/(15+18

$\frac{x}{15} = \frac{x + 20}{33}$

$33 x = 15 x + 300$

$18 x = 300$

$x = 16 \frac{2}{3}$

Mar 30, 2018

$C$

#### Explanation:

We can use the Side-Splitter Theorem to solve this problem. It states:

• If a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.

Since $O P$ || $L N$, this theorem applies.

So we can set up this proportion:
$\frac{x}{20} = \frac{15}{18}$

Now cross multiply and solve:
$\frac{x}{20} = \frac{15}{18}$

$x \times 18 = 20 \times 15$

$18 x = 300$

$x = \frac{300}{18} \rightarrow 16 \frac{12}{18} \rightarrow 16 \frac{2}{3}$

So the answer is $C$

Mar 30, 2018

Answer: $x = 16 \cdot \frac{2}{3}$

#### Explanation:

Since $O P$ is parallel to $L N$, we know that $\angle M O P = \angle M L N$ and $\angle M P O = \angle M N L$ from the Corresponding Angles Theorem

Further, we also have that $\angle O M P = \angle L M N$ since they are the same angle.

Therefore $\triangle O M P$ is similar to $\triangle L M N$ (triangleOMP~triangleLMN)

Since similar triangles have the same side length ratio:
$\frac{M O}{M L} = \frac{M P}{M N}$

Plugging numbers in, we have:
$\frac{x}{x + 20} = \frac{15}{15 + 18}$

We can now solve this equation by cross multiplication:
$33 x = 15 \left(x + 20\right)$
$33 x = 15 x + 300$
$18 x = 300$
$x = 16 \cdot \frac{2}{3}$