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

two triangles, left triangle labeled w, v, u with angles at w: 60°, at v: 63°, sides wv=43, vu=45, uw=46; right triangle labeled x, y, z with angles at y: 63°, at x: 57°, sides yz=34.4, xy=36, zx=x

Question

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

two triangles, left triangle labeled w, v, u with angles at w: 60°, at v: 63°, sides wv=43, vu=45, uw=46; right triangle labeled x, y, z with angles at y: 63°, at x: 57°, sides yz=34.4, xy=36, zx=x

Accepted Answer

# Explanation:
## Step1: Check Triangle Similarity
First, we check if $\triangle VWU$ and $\triangle XYZ$ are similar. Let's find the angles of $\triangle VWU$. The sum of angles in a triangle is $180^\circ$. So, $\angle U = 180^\circ - 60^\circ - 63^\circ = 57^\circ$. In $\triangle XYZ$, the angles are $63^\circ$ (at $Y$), $57^\circ$ (at $X$), so the third angle $\angle Z = 180^\circ - 63^\circ - 57^\circ = 60^\circ$. So, $\triangle VWU \sim \triangle XYZ$ by AA similarity (since corresponding angles are equal: $60^\circ$, $63^\circ$, $57^\circ$).

## Step2: Determine Corresponding Sides
Now, we need to find the corresponding sides. In $\triangle VWU$, sides: $VW = 43$, $WU = 46$, $VU = 45$. In $\triangle XYZ$, sides: $YZ = 34.4$, $XY = 36$, $XZ = x$. Let's find the ratio of similarity. Let's match the angles: $\angle V = 63^\circ$ corresponds to $\angle Y = 63^\circ$, $\angle W = 60^\circ$ corresponds to $\angle Z = 60^\circ$, $\angle U = 57^\circ$ corresponds to $\angle X = 57^\circ$. So, side $VU$ (opposite $\angle W = 60^\circ$) in $\triangle VWU$ is $45$, and side $YZ$ (opposite $\angle X = 57^\circ$)? Wait, maybe better to match sides by angle correspondence. Wait, $\angle V = 63^\circ$ (side $WU = 46$ opposite), $\angle Y = 63^\circ$ (side $XZ = x$ opposite). $\angle W = 60^\circ$ (side $VU = 45$ opposite), $\angle Z = 60^\circ$ (side $XY = 36$ opposite). $\angle U = 57^\circ$ (side $VW = 43$ opposite), $\angle X = 57^\circ$ (side $YZ = 34.4$ opposite). Let's check the ratio. Let's take the side opposite $57^\circ$: in $\triangle VWU$, side opposite $57^\circ$ (which is $\angle U$) is $VW = 43$. In $\triangle XYZ$, side opposite $57^\circ$ (which is $\angle X$) is $YZ = 34.4$. So the ratio of similarity $k = \frac{34.4}{43} = 0.8$.

## Step3: Find Other Sides of $\triangle XYZ$
Now, find $x$ (which is $XZ$, opposite $\angle Y = 63^\circ$). In $\triangle VWU$, the side opposite $\angle Y = 63^\circ$ (which is $\angle V = 63^\circ$) is $WU = 46$. So, using the similarity ratio $k = 0.8$, we have $x = 46 \times k = 46 \times 0.8 = 36.8$? Wait, no, wait. Wait, the side $YZ = 34.4$ corresponds to which side in $\triangle VWU$? Let's see, $\angle U = 57^\circ$, side $VW = 43$ (opposite $\angle U$). $\angle X = 57^\circ$, side $YZ = 34.4$ (opposite $\angle X$). So the ratio is $\frac{YZ}{VW} = \frac{34.4}{43} = 0.8$. So that's the scale factor. Now, side $XY = 36$ in $\triangle XYZ$ corresponds to which side in $\triangle VWU$? $\angle Z = 60^\circ$, side $XY = 36$ (opposite $\angle Z$). In $\triangle VWU$, $\angle W = 60^\circ$, side $VU = 45$ (opposite $\angle W$). Let's check the ratio: $\frac{XY}{VU} = \frac{36}{45} = 0.8$, which matches the previous ratio. Good, so the scale factor is $0.8$. Now, side $XZ = x$ in $\triangle XYZ$ corresponds to side $WU = 46$ in $\triangle VWU$ (since $\angle Y = 63^\circ$ in $\triangle XYZ$ corresponds to $\angle V = 63^\circ$ in $\triangle VWU$, and side opposite $\angle V$ is $WU = 46$, side opposite $\angle Y$ is $XZ = x$). So, $x = 46 \times 0.8 = 36.8$? Wait, no, wait: if the scale factor is from $\triangle VWU$ to $\triangle XYZ$, then $XYZ$ is smaller, so $x = 46 \times 0.8 = 36.8$? Wait, but let's check with another side. Wait, $YZ = 34.4$ is corresponding to $VW = 43$, since $\angle U = 57^\circ$ (side $VW = 43$ opposite) and $\angle X = 57^\circ$ (side $YZ = 34.4$ opposite). So, $\frac{YZ}{VW} = \frac{34.4}{43} = 0.8$, which is correct. So, all ratios are $0.8$. So, now, the sides of $\triangle XYZ$ are $YZ = 34.4$, $XY = 36$, $XZ = x = 46 \times 0.8 = 36.8$? Wait, no, wait: $WU = 46$ (side in $\triangle VWU$) corresponds to $XZ = x$ (side in $\triangle XYZ$) because $\angle V = 63^\circ$ (opposite $WU$) and $\angle Y = 63^\circ$ (opposite $XZ$). So, since the scale factor is $0.8$ (because $34.4 / 43 = 0.8$, $36 / 45 = 0.8$), then $x = 46 \times 0.8 = 36.8$. Wait, but let's confirm the perimeter. First, find all sides of $\triangle XYZ$: $YZ = 34.4$, $XY = 36$, $XZ = 36.8$? Wait, no, that can't be, because $34.4 + 36 + 36.8 = 107.2$, but let's check the original triangle's perimeter: $43 + 46 + 45 = 134$. Then, $134 \times 0.8 = 107.2$, which matches. Wait, but let's check the sides again. Wait, maybe I mixed up the correspondence. Let's list the angles and opposite sides:

In $\triangle VWU$:
- $\angle V = 63^\circ$, opposite side $WU = 46$
- $\angle W = 60^\circ$, opposite side $VU = 45$
- $\angle U = 57^\circ$, opposite side $VW = 43$

In $\triangle XYZ$:
- $\angle Y = 63^\circ$, opposite side $XZ = x$
- $\angle Z = 60^\circ$, opposite side $XY = 36$
- $\angle X = 57^\circ$, opposite side $YZ = 34.4$

So, the ratio of similarity is $\frac{\text{side in } XYZ}{\text{side in } VWU}$. So, for $\angle X = 57^\circ$, side $YZ = 34.4$ (in $XYZ$) corresponds to side $VW = 43$ (in $VWU$): $\frac{34.4}{43} = 0.8$. For $\angle Z = 60^\circ$, side $XY = 36$ (in $XYZ$) corresponds to side $VU = 45$ (in $VWU$): $\frac{36}{45} = 0.8$. For $\angle Y = 63^\circ$, side $XZ = x$ (in $XYZ$) corresponds to side $WU = 46$ (in $VWU$): $\frac{x}{46} = 0.8$, so $x = 46 \times 0.8 = 36.8$. Correct.

## Step3: Calculate Perimeter of $\triangle XYZ$
Now, the perimeter of $\triangle XYZ$ is $YZ + XY + XZ = 34.4 + 36 + 36.8$. Let's calculate that: $34.4 + 36 = 70.4$, $70.4 + 36.8 = 107.2$. Wait, but let's check with the scale factor. The perimeter of $\triangle VWU$ is $43 + 46 + 45 = 134$. Since the scale factor is $0.8$, the perimeter of $XYZ$ should be $134 \times 0.8 = 107.2$, which matches. So that's correct.

# Answer:
The perimeter of $\triangle XYZ$ is $\boxed{107.2}$.

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QUESTION IMAGE

Question

Explanation:

Step1: Check Triangle Similarity

Step2: Determine Corresponding Sides

Step3: Find Other Sides of \(\triangle XYZ\)

Answer: