To determine the geometric figure formed by the points $C(4,6)$, $A(3,4)$, $R(4,2)$, and $E(5,4)$, and to find its area, we can follow these steps:

Step 1: Plot the Points

1. $C(4,6)$
2. $A(3,4)$
3. $R(4,2)$
4. $E(5,4)$

Step 2: Connect the Points

• Connect $C$ to $A$ to form segment $CA$.
• Connect $A$ to $R$ to form segment $AR$.
• Connect $R$ to $E$ to form segment $RE$.
• Connect $E$ to $C$ to form segment $CE$.

Step 3: Identify the Geometric Figure

After connecting the points, we observe that the figure formed is a quadrilateral.

Step 4: Calculate the Area of the Quadrilateral

To find the area of the quadrilateral, we can use the Shoelace Theorem (or Gauss's area formula for polygons). The formula is given by:

$\text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n-1}(x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|$

For our quadrilateral with vertices $(x_1, y_1) = (4,6)$, $(x_2, y_2) = (3,4)$, $(x_3, y_3) = (4,2)$, $(x_4, y_4) = (5,4)$, the area calculation is:

\text{Area} &= \frac{1}{2} \left| (4 \cdot 4 + 3 \cdot 2 + 4 \cdot 4 + 5 \cdot 6) - (6 \cdot 3 + 4 \cdot 4 + 2 \cdot 5 + 4 \cdot 4) \right| \\ &= \frac{1}{2} \left| (16 + 6 + 16 + 30) - (18 + 16 + 10 + 16) \right| \\ &= \frac{1}{2} \left| 68 - 60 \right| \\ &= \frac{1}{2} \left| 8 \right| \\ &= 4 \end{aligned}$$ ### Conclusion The geometric figure formed is a quadrilateral, and the area of this quadrilateral is $$4$$ square units. Would you like any more details or have any questions? Here are five related questions to expand your understanding: 1. How does the Shoelace Theorem work for polygons with more than four sides? 2. What are some properties of quadrilaterals that can be used to verify their type? 3. How would you determine if the given quadrilateral is a special type like a rectangle or a trapezoid? 4. What other methods can be used to find the area of a quadrilateral if the coordinates are known? 5. How does changing one of the vertices affect the area of the quadrilateral? **Tip:** When plotting points and connecting them, always ensure the vertices are listed in consecutive order to accurately determine the shape and calculate the area correctly.