To calculate $ED$ in the given figure, we need to analyze the triangle or quadrilateral involving segment $ED$ and apply appropriate geometric properties or formulas. Here's the analysis based on the diagram:

Step 1: Breakdown of Information

1. Coordinates and dimensions available:

   • $AB = 3$, $BC = 4$, and $AC = 5$ (a right triangle by the Pythagorean theorem).
   • $AD = 3.1623$, and $CD = 7.8102$.

2. Angle at $C$: $\angle DCE = 71.5651^\circ$.

3. $\triangle CDE$ is the key triangle for finding $ED$, given the dimensions $CD = 7.8102$ and $\angle DCE = 71.5651^\circ$.

Step 2: Apply the Law of Cosines

The Law of Cosines for any triangle states: $c^2 = a^2 + b^2 - 2ab\cos(\theta),$ where:

• $c$ is the side opposite the angle,
• $a$ and $b$ are the other two sides,
• $\theta$ is the angle between $a$ and $b$.

In $\triangle CDE$:

• $a = CD = 7.8102$,
• $b = DE$ (to be found),
• $\theta = \angle DCE = 71.5651^\circ$.

Step 3: Calculate $ED$

We rearrange the Law of Cosines to solve for $DE$: $DE = \sqrt{CD^2 + CE^2 - 2 \cdot CD \cdot CE \cdot \cos(\angle DCE)}.$

Plugging in the values:

• $CD = 7.8102$,
• $CE = 3.1623$,
• $\cos(71.5651^\circ) \approx 0.32557$ (from trigonometric tables).

$DE = \sqrt{(7.8102)^2 + (3.1623)^2 - 2(7.8102)(3.1623)(0.32557)}.$

Step 4: Simplify the Expression

1. Square the values:

   • $CD^2 = 7.8102^2 = 60.992$,
   • $CE^2 = 3.1623^2 = 10.001$.

2. Multiply $2 \cdot CD \cdot CE \cdot \cos(\angle DCE)$:

   • $2 \cdot 7.8102 \cdot 3.1623 \cdot 0.32557 \approx 16.036$.

3. Substitute into the formula: $DE = \sqrt{60.992 + 10.001 - 16.036}.$

4. Simplify further: $DE = \sqrt{54.957} \approx 7.41.$

Final Answer:

$ED \approx 7.41 \, \text{units}.$

Formula or Property Used:

1. Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(\theta)$.
2. Calculations rely on trigonometric properties and precise angle measures.

Would you like more details or explanations? 😊

Related Questions:

1. What are other ways to find $DE$ without using the Law of Cosines?
2. How is the cosine of $\angle DCE = 71.5651^\circ$ determined?
3. How does the Law of Sines compare to the Law of Cosines for solving this triangle?
4. Can the area of $\triangle CDE$ be calculated using the given information?
5. What trigonometric relationships are essential for understanding triangles like $\triangle CDE$?

Tip:

Always verify if simpler methods, like using Pythagoras or right-angle triangle properties, apply before moving to advanced formulas!