To find the perimeter of triangle EFG, we need to determine the lengths of all three sides of the triangle first.
From the given information, we have three sides with their respective lengths:
- EF = 55
- FG = x (unknown)
- EG = 27.5
The properties of a triangle dictate that the sum of the lengths of any two sides must be greater than the length of the third side.
We also have the angles of triangle EFG:
- Angle E = 53°
- Angle F = 59°
- Angle G = 68°
While we don’t have the full set of dimensions given, we can calculate the unknown side length FG using the Law of Sines:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Where:
- a, b, c are the lengths of the sides opposite to angles A, B, C respectively.
- A, B, C are the angles at the vertices of the triangle.
In our case:
- \( a = 55 \) (EF)
- \( b = 27.5 \) (EG)
- \( c = x \) (FG)
- \( A = 53° \) (angle at vertex E)
- \( B = 59° \) (angle at vertex F)
- \( C = 68° \) (angle at vertex G)
Now, we can set up our equation using the Law of Sines: \[ \frac{EG}{\sin(\angle F)} = \frac{EF}{\sin(\angle G)} \] This translates to: \[ \frac{27.5}{\sin(59°)} = \frac{55}{\sin(68°)} \]
Now we can find \( FG \) using the Law of Sines:
From the Law of Sines for \( FG \): \[ \frac{x}{\sin(53°)} = \frac{27.5}{\sin(59°)} \]
Rearranging gives: \[ x = \frac{27.5 \cdot \sin(53°)}{\sin(59°)} \]
Calculating \( x \):
- First, compute \( \sin(53°) \) and \( \sin(59°) \):
Using a calculator:
- \( \sin(53°) \approx 0.7986 \)
- \( \sin(59°) \approx 0.8572 \)
Now substituting these into the formula: \[ x = \frac{27.5 \cdot 0.7986}{0.8572} \] Calculating gives: \[ x \approx \frac{21.9465}{0.8572} \approx 25.6 \]
Now, we have:
- \( EF = 55 \)
- \( EG = 27.5 \)
- \( FG \approx 25.6 \)
Now, we can find the perimeter:
\[ \text{Perimeter} = EF + FG + EG = 55 + 25.6 + 27.5 \] Calculating that gives: \[ \text{Perimeter} \approx 55 + 25.6 + 27.5 \approx 108.1 \]
So, the perimeter of triangle EFG is approximately 108.1 units when rounded to the nearest tenth.