find the perimeter of $\triangle uvw$. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

(image of two triangles: left triangle $rst$ with $rs=20$, $st=15$, $rt=27$, angles $\angle r=34^\circ$, $\angle t=49^\circ$, $\angle s=97^\circ$; right triangle $uvw$ with $wv=37.5$, $vu=50$, angle $\angle w=49^\circ$, $\angle v=97^\circ$, $\angle u=34^\circ$, side $wu=x$)

Question

find the perimeter of $\triangle uvw$. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

(image of two triangles: left triangle $rst$ with $rs=20$, $st=15$, $rt=27$, angles $\angle r=34^\circ$, $\angle t=49^\circ$, $\angle s=97^\circ$; right triangle $uvw$ with $wv=37.5$, $vu=50$, angle $\angle w=49^\circ$, $\angle v=97^\circ$, $\angle u=34^\circ$, side $wu=x$)

Accepted Answer

# Explanation:
## Step1: Identify Similar Triangles
Triangles $ \triangle RST $ and $ \triangle UVW $ have the same angle measures ($ 34^\circ $, $ 49^\circ $, $ 97^\circ $), so they are similar by AA (Angle - Angle) similarity criterion.

## Step2: Find the Scale Factor
In $ \triangle RST $, the sides are $ 20 $, $ 15 $, and $ 27 $. In $ \triangle UVW $, one of the corresponding sides (to $ 15 $ in $ \triangle RST $) is $ 50 $. The scale factor $ k $ is $ \frac{50}{15}=\frac{10}{3}$.

## Step3: Find the Length of $ x $
The side of length $ 27 $ in $ \triangle RST $ corresponds to side $ x $ in $ \triangle UVW $. Using the scale factor, $ x = 27\times\frac{10}{3}=90 $.

## Step4: Find the Perimeter of $ \triangle RST $
Perimeter of $ \triangle RST=20 + 15+27 = 62 $.

## Step5: Find the Perimeter of $ \triangle UVW $
Since the ratio of perimeters of similar triangles is equal to the scale factor, let $ P $ be the perimeter of $ \triangle UVW $. We know that $ \frac{P}{62}=\frac{10}{3} $, so $ P = 62\times\frac{10}{3}=\frac{620}{3}\approx206.7 $? Wait, no, wait. Wait, we can also calculate the perimeter by finding all sides of $ \triangle UVW $.

Wait, another way: The side of length $ 20 $ in $ \triangle RST $ corresponds to the side of length $ 37.5 $ in $ \triangle UVW $? Wait, no, let's re - check the correspondence.

Wait, in $ \triangle RST $: angles are $ \angle R = 34^\circ $, $ \angle T=49^\circ $, $ \angle S = 97^\circ $. In $ \triangle UVW $: $ \angle U = 34^\circ $, $ \angle W = 49^\circ $, $ \angle V=97^\circ $. So the correspondence is $ R
ightarrow U $, $ S
ightarrow V $, $ T
ightarrow W $.

So side $ RS = 20 $ (opposite $ \angle T = 49^\circ $) corresponds to $ UV = 50 $ (opposite $ \angle W = 49^\circ $)? Wait, no, $ \angle T = 49^\circ $, side opposite $ \angle T $ is $ RS = 20 $. In $ \triangle UVW $, $ \angle W = 49^\circ $, side opposite $ \angle W $ is $ UV = 50 $. So the ratio of sides is $ \frac{UV}{RS}=\frac{50}{20}=\frac{5}{2} $.

Ah, I made a mistake in the correspondence earlier. Let's correct that.

So $ \triangle RST $ has sides: $ RS = 20 $ (opposite $ \angle T = 49^\circ $), $ ST = 15 $ (opposite $ \angle R = 34^\circ $), $ RT = 27 $ (opposite $ \angle S = 97^\circ $).

In $ \triangle UVW $: $ UV = 50 $ (opposite $ \angle W = 49^\circ $), $ VW = 37.5 $ (opposite $ \angle U = 34^\circ $), $ UW=x $ (opposite $ \angle V = 97^\circ $).

So the ratio of sides: $ \frac{UV}{RS}=\frac{50}{20}=\frac{5}{2} $, $ \frac{VW}{ST}=\frac{37.5}{15}=\frac{5}{2} $, so the scale factor is $ \frac{5}{2} $.

Then, $ \frac{UW}{RT}=\frac{5}{2} $, so $ UW=x=\frac{5}{2}\times27 = 67.5 $? Wait, no, $ RT = 27 $ (opposite $ \angle S = 97^\circ $), $ UW $ is opposite $ \angle V = 97^\circ $, so $ x=\frac{5}{2}\times27 = 67.5 $? Wait, now I am confused. Let's use the correct correspondence.

Let's list the angles:

In $ \triangle RST $:

$ \angle R = 34^\circ $, $ \angle S=97^\circ $, $ \angle T = 49^\circ $

In $ \triangle UVW $:

$ \angle U = 34^\circ $, $ \angle V = 97^\circ $, $ \angle W = 49^\circ $

So the correspondence is $ R
ightarrow U $, $ S
ightarrow V $, $ T
ightarrow W $

So side $ RS $ (between $ \angle R $ and $ \angle S $): length $ 20 $

Side $ UV $ (between $ \angle U $ and $ \angle V $): length $ 50 $

Side $ ST $ (between $ \angle S $ and $ \angle T $): length $ 15 $

Side $ VW $ (between $ \angle V $ and $ \angle W $): length $ 37.5 $

Side $ RT $ (between $ \angle R $ and $ \angle T $): length $ 27 $

Side $ UW $ (between $ \angle U $ and $ \angle W $): length $ x $

The ratio of $ UV $ to $ RS $ is $ \frac{50}{20}=\frac{5}{2} $, the ratio of $ VW $ to $ ST $ is $ \frac{37.5}{15}=\frac{5}{2} $, so the scale factor is $ \frac{5}{2} $

Then the length of $ UW $ (corresponding to $ RT $) is $ 27\times\frac{5}{2}=67.5 $

Now, perimeter of $ \triangle UVW=37.5 + 50+67.5=155 $

Wait, let's check:

Perimeter of $ \triangle RST=20 + 15 + 27=62 $

Perimeter of $ \triangle UVW $: using scale factor $ \frac{5}{2} $, $ 62\times\frac{5}{2}=155 $. Yes, that's correct. Because $ 20\times\frac{5}{2}=50 $? No, wait $ 20\times\frac{5}{2}=50 $? No, $ 20\times\frac{5}{2}=50 $ is wrong, $ 20\times\frac{5}{2}=50 $? Wait $ 20\times2.5 = 50 $, yes. $ 15\times2.5=37.5 $, $ 27\times2.5 = 67.5 $. Then $ 37.5+50 + 67.5=155 $.

# Answer:
The perimeter of $ \triangle UVW $ is $ \boldsymbol{155} $.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Identify Similar Triangles

Step2: Find the Scale Factor

Step3: Find the Length of \( x \)

Step4: Find the Perimeter of \( \triangle RST \)

Step5: Find the Perimeter of \( \triangle UVW \)

Answer: