c) Calculate the length of each diagonal of $$ A B C D $$. Based on this result and your results in (a) and (b), what type of special quadrilateral is $$ A B C D $$ ? Length of $$ \overline{A C} $$ : Length of $$ \overline{B D} $$ :

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Certainly! To determine the lengths of the diagonals of quadrilateral $$ A B C D $$ and identify its type based on your previous results from parts (a) and (b), follow the steps below. If you have specific coordinates or side lengths from parts (a) and (b), you can substitute them accordingly.

### **1. Calculating the Lengths of the Diagonals**

Assuming you have the coordinates of the vertices $$ A $$, $$ B $$, $$ C $$, and $$ D $$, you can use the **Distance Formula** to calculate the lengths of the diagonals $$ \overline{A C} $$ and $$ \overline{B D} $$.

The **Distance Formula** between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

**a) Length of $$ \overline{A C} $$:**

Let’s say:
- Point $$ A $$ has coordinates $$ (x_A, y_A) $$
- Point $$ C $$ has coordinates $$ (x_C, y_C) $$

Then,

$$
\text{Length of } \overline{A C} = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2}
$$

**b) Length of $$ \overline{B D} $$:**

Let’s say:
- Point $$ B $$ has coordinates $$ (x_B, y_B) $$
- Point $$ D $$ has coordinates $$ (x_D, y_D) $$

Then,

$$
\text{Length of } \overline{B D} = \sqrt{(x_D - x_B)^2 + (y_D - y_B)^2}
$$

**_Example Calculation:_**

*Suppose you have the following coordinates:*
- $$ A(1, 2) $$
- $$ B(4, 6) $$
- $$ C(7, 2) $$
- $$ D(4, -2) $$

**Calculate $$ \overline{A C} $$:**

$$
\overline{A C} = \sqrt{(7 - 1)^2 + (2 - 2)^2} = \sqrt{6^2 + 0^2} = \sqrt{36} = 6
$$

**Calculate $$ \overline{B D} $$:**

$$
\overline{B D} = \sqrt{(4 - 4)^2 + (-2 - 6)^2} = \sqrt{0^2 + (-8)^2} = \sqrt{64} = 8
$$

*So,*
- **Length of $$ \overline{A C} $$:** 6 units
- **Length of $$ \overline{B D} $$:** 8 units

### **2. Determining the Type of Quadrilateral**

Based on the lengths of the diagonals and your previous findings in parts (a) and (b), you can identify the type of quadrilateral $$ A B C D $$ is. Here's how different properties can help:

- **Parallelogram:** Both pairs of opposite sides are equal and parallel. Diagonals bisect each other but are not necessarily equal.
  
- **Rectangle:** All properties of a parallelogram, and the diagonals are equal in length.
  
- **Rhombus:** All sides are equal, diagonals bisect each other at right angles, but diagonals are not necessarily equal.
  
- **Square:** Combines the properties of a rectangle and a rhombus. All sides are equal, and diagonals are equal and bisect each other at right angles.
  
- **Kite:** Two distinct pairs of adjacent sides are equal. One diagonal bisects the other at right angles.

**_Using the Example:_**

With diagonals of lengths 6 and 8 units:
- **Not a Rectangle or Square:** Since the diagonals are not equal.
  
- **Not a Generic Parallelogram:** Unless your parts (a) and (b) confirm that opposite sides are equal and parallel.
  
- **Possibly a Rhombus:** If all sides are equal **and** diagonals bisect each other at right angles.
  
- **Possibly a Kite:** If two distinct pairs of adjacent sides are equal **and** one diagonal bisects the other at right angles.

**Conclusion Based on the Example:**

If, from parts (a) and (b), you found that all sides are equal and the diagonals intersect at right angles, then $$ A B C D $$ is a **Square**. If only the sides are equal without the diagonals being equal, it would be a **Rhombus**. If only one pair of opposite sides is equal and the diagonals are of different lengths without right angles, it could be a **Parallelogram**.

---

**_Please provide the specific coordinates or additional information from parts (a) and (b) if you need a more precise identification of the quadrilateral._**
To determine the lengths of the diagonals $$ \overline{A C} $$ and $$ \overline{B D} $$ of quadrilateral $$ A B C D $$, use the Distance Formula: $$ \text{Length of } \overline{A C} = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} $$ $$ \text{Length of } \overline{B D} = \sqrt{(x_D - x_B)^2 + (y_D - y_B)^2} $$ After calculating, compare the lengths to identify the type of quadrilateral: - **Square:** Both diagonals are equal and bisect each other at right angles. - **Rhombus:** All sides are equal, diagonals bisect each other but are not necessarily equal. - **Rectangle:** Diagonals are equal but not necessarily bisecting at right angles. - **Parallelogram:** Opposite sides are equal and parallel, diagonals bisect each other. - **Kite:** Two pairs of adjacent sides are equal, one diagonal bisects the other at right angles. Ensure that the specific lengths and properties from parts (a) and (b) are used to accurately determine the type of quadrilateral.

c) Calculate the length of each diagonal of \( A B C D \). Based on this result and your results in (a) and (b), what type of special quadrilateral is \( A B C D \) ? Length of \( \overline{A C} \) : Length of \( \overline{B D} \) :

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