To solve for $x$, we analyze the given triangle $\triangle DEF$, subdivided into smaller triangles by segment $GH$. Here's a step-by-step approach:

1. Note the Geometry: $\triangle DEF$ is split into two smaller triangles:

   • $\triangle FGH$, with side $FG = 6$, $FH = 4$, and $\angle GHF = 82^\circ$.
   • $\triangle DGE$, sharing $GH$, with $DE = 4.8$ and $\angle DGE = 82^\circ$.

2. Find $GH$ using the Law of Cosines in $\triangle FGH$:
   Using the formula:
   $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$ where $c = GH$, $a = FG = 6$, $b = FH = 4$, and $\angle GHF = 82^\circ$:
   $GH^2 = 6^2 + 4^2 - 2(6)(4)\cos(82^\circ)$ Calculate this to find $GH$.

3. Use $GH$ to Find $x = DE$:
   In $\triangle DGE$, use the Law of Cosines again: $x^2 = GH^2 + 4.8^2 - 2(GH)(4.8)\cos(82^\circ)$ Substitute $GH$ (calculated in Step 2) to solve for $x$.

Let me calculate this systematically.import math

Given data

FG = 6 FH = 4 angle_GHF = math.radians(82) # Convert angle to radians DE = 4.8 angle_DGE = math.radians(82) # Convert angle to radians

Step 1: Calculate GH using the Law of Cosines in triangle FGH

GH_squared = FG2 + FH2 - 2 * FG * FH * math.cos(angle_GHF) GH = math.sqrt(GH_squared)

Step 2: Use GH to calculate x using the Law of Cosines in triangle DGE

x_squared = GH2 + DE2 - 2 * GH * DE * math.cos(angle_DGE) x = math.sqrt(x_squared)

GH, x(6.731990133230801, 7.704890427914184)(6.731990133230801, 7.704890427914184)successThe length of $GH$ is approximately $6.73$, and the value of $x$ (the length of $DE$) is approximately $7.70$.

Would you like a detailed explanation of the calculations or any further assistance?

Here are 5 related questions for deeper understanding:

1. How does the Law of Cosines relate to solving non-right triangles?
2. Can you derive the Law of Cosines step by step?
3. How would the result change if one of the angles was different?
4. Can you verify the result using the Law of Sines instead?
5. What other methods could you use to find $x$ in this triangle?

Tip: Always double-check angle measurements to ensure they are consistent in radians or degrees during calculations.