Question

Find the perimeter of triangle, W, X, Y△WXY. Round your answer to the nearest tenth if necessary. Figures are not necessarily drawn to scale.
T
U
V
W
X
Y
54°
38°
54°
88°
3
5
6
x
4
4.8
Answer
Attempt 1 out of 2

Answer 1

ok if wxy already has a side with the length of 6 and another saide of 4.8 what is the third side

Answer 2

To find the perimeter of triangle \( \triangle WXY \), we need to determine the lengths of all sides \( WX \), \( XY \), and \( YW \).

From the information given:

- We have angles: \( \angle W = 54^\circ \), \( \angle X = 38^\circ \), \( \angle Y = 88^\circ \)
- We also have sides provided: \( WX = 3 \), \( XY = 5 \), and \( YW = 6 \)

### Step 1: Calculate the Unknown Side \( YW \)
Since we have the lengths of two sides and the angles opposite these sides, we can utilize the Law of Sines to find the third side.

The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Where:
- \( a, b, c \) are the sides of the triangle
- \( A, B, C \) are the angles opposite these sides

Here:
- \( a = YW \)
- \( b = 5 \) (opposite \( 54^\circ \))
- \( c = 3 \) (opposite \( 38^\circ \))
- \( A = 54^\circ \)
- \( B = 38^\circ \)
- \( C = 88^\circ \)

### Step 2: Use the Law of Sines
Using the two known sides:

\[
\frac{YW}{\sin(38^\circ)} = \frac{5}{\sin(54^\circ)}
\]

Now plug in the values:

\[
YW = \frac{5 \cdot \sin(38^\circ)}{\sin(54^\circ)}
\]

### Step 3: Calculate the Values
Using values of \( \sin(38^\circ) \) and \( \sin(54^\circ) \):
- \( \sin(38^\circ) \approx 0.6157 \)
- \( \sin(54^\circ) \approx 0.8090 \)

Now substitute these values in:

\[
YW = \frac{5 \cdot 0.6157}{0.8090} \approx \frac{3.0785}{0.8090} \approx 3.804
\]

### Step 4: Calculate the Perimeter
Now we can calculate the perimeter \( P \) of triangle \( WXY \):

\[
P = WX + XY + YW = 3 + 5 + 3.804 \approx 11.804
\]

### Step 5: Round the Final Answer
Finally, round to the nearest tenth:

\[
P \approx 11.8
\]

Therefore, the perimeter of triangle \( \triangle WXY \) is **11.8**.

Answer 3

If triangle \( WXY \) already has two sides with lengths of \( 6 \) and \( 4.8 \), we can use these lengths to find the length of the third side, but we need to clarify if you want to find the length of the third side using another triangle's side length or if the triangle is already defined by these lengths.

### Given:
- Side \( WX = 6 \)
- Side \( XY = 4.8 \)
- Let's denote the unknown side \( YW \) (the third side).

### Step 1: Check if this is a triangle
In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the triangle inequality theorem.

Using the sides defined:
1. \( WX + XY > YW \) implies \( 6 + 4.8 > YW \) or \( YW < 10.8 \)
2. \( WX + YW > XY \) implies \( 6 + YW > 4.8 \) or \( YW > -1.2 \) (always true for positive side lengths)
3. \( XY + YW > WX \) implies \( 4.8 + YW > 6 \) or \( YW > 1.2 \)

From this, we conclude:
- \( 1.2 < YW < 10.8 \)

### Conclusion
Since we don't have length for \( YW \) explicitly stated, it can take any value in the range \( 1.2 < YW < 10.8 \) to form a valid triangle.

If you have a specific third side or requirements regarding \( YW \) based on angles or a specific triangle's properties, please clarify! Otherwise, without additional information, the third side of triangle \( WXY \) could be any length between \( 1.2 \) and \( 10.8 \) while ensuring a triangle can be formed.