Question

question
in $\triangle tvu$ shown below, point $w$ is on $\overline{uv}$, and point $x$ is on $\overline{tv}$ so that $\angle tuv \cong \angle wxv$. if $vt = 45$, $vw = 18$, and $tu = 35$, find the length of $wx$. figures are not necessarily drawn to scale.
(image of triangle $tvu$ with $w$ on $uv$, $x$ on $tv$, and segment $wx$ drawn)

Explanation:

Step1: Identify Similar Triangles

Since $\angle TUV \cong \angle WXV$ and $\angle V$ is common to both $\triangle TUV$ and $\triangle WXV$, by the AA (Angle - Angle) similarity criterion, $\triangle TUV \sim \triangle WXV$.

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So we have the proportion:
$\frac{VW}{VU}=\frac{WX}{TU}$
First, we need to find the length of $VU$. We know that $VW = 18$ and $WU$ (wait, actually $VU=VW + WU$? Wait, no, looking at the diagram, $W$ is on $UV$, so $UV=UW + WV$? Wait, no, the problem says $W$ is on $\overline{UV}$, so $UV = UW+WV$? Wait, no, the given lengths: $WT = 45$? Wait, no, the problem says $VT = 45$? Wait, the problem statement: "In $\triangle TVU$ shown below, point $W$ is on $\overline{UV}$, and point $X$ is on $\overline{TV}$ so that $\angle TUV\cong\angle WXV$. If $VT = 45$, $VW = 18$, and $TU = 35$, find the length of $WX$." Wait, maybe I misread. Let's re - read: "In $\triangle TVU$... point $W$ is on $\overline{UV}$, and point $X$ is on $\overline{TV}$ so that $\angle TUV\cong\angle WXV$. If $VT = 45$, $VW = 18$, and $TU = 35$, find the length of $WX$."

Wait, actually, since $\triangle TUV\sim\triangle WXV$ (AA similarity: $\angle TUV=\angle WXV$ and $\angle V=\angle V$), the ratio of corresponding sides:
$\frac{WX}{TU}=\frac{VW}{VT}$

Wait, maybe I mixed up the sides. Let's correct: In $\triangle TUV$ and $\triangle WXV$:

• $\angle TUV=\angle WXV$ (given)
• $\angle V=\angle V$ (common angle)

So the correspondence is $\triangle TUV\sim\triangle WXV$ (order of angles: $\angle TUV\cong\angle WXV$, $\angle UTV\cong\angle XWV$, $\angle V\cong\angle V$). So the corresponding sides: $TU$ corresponds to $WX$, $TV$ corresponds to $WV$? No, wait, $TV$ is the side from $T$ to $V$, length $VT = 45$. $WV$ is the side from $W$ to $V$, length $VW = 18$. $TU$ is the side from $T$ to $U$, length $TU = 35$. And $WX$ is the side from $W$ to $X$ which we need to find.

So the ratio of similarity is $\frac{VW}{VT}=\frac{WX}{TU}$

Step3: Substitute Values and Solve

We know that $VW = 18$, $VT = 45$, and $TU = 35$. Substitute these values into the proportion:
$\frac{18}{45}=\frac{WX}{35}$

To solve for $WX$, we can cross - multiply:
$45\times WX=18\times35$

First, calculate $18\times35 = 630$

Then, $WX=\frac{630}{45}$

Simplify $\frac{630}{45}$: divide numerator and denominator by 15, we get $\frac{42}{3}=14$

Answer:

14