Question

Given the vertices, determine the most specific classification: Quadrilateral, Parallelogram, Rectangle, Rhombus, or Square. Make sure to show ALL of your work that justifies your answer. Remember you may be finding the distances, slopes or midpoints so make sure to show ALL of your work!

COORDINATE PROOF

V(-8, -6), A(8. 2), S(10, -2), 7(-6, -10)

GRAPH

Summary: I am a _______ because_______

Answer 1

Show work

Answer 2

To classify the quadrilateral formed by the points V(-8, -6), A(8, 2), S(10, -2), and 7(-6, -10), we will analyze the vertices by calculating the lengths of the sides, the slopes of the sides, and the diagonals where necessary.

### Step 1: Calculate the lengths of the sides

To find the lengths of the sides, we will use the distance formula:

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

#### Side VA
\[
VA = \sqrt{(8 - (-8))^2 + (2 - (-6))^2} = \sqrt{(8 + 8)^2 + (2 + 6)^2} = \sqrt{(16)^2 + (8)^2} = \sqrt{256 + 64} = \sqrt{320} = 8\sqrt{5}
\]

#### Side AS
\[
AS = \sqrt{(10 - 8)^2 + (-2 - 2)^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

#### Side SV
\[
SV = \sqrt{(-8 - 10)^2 + (-6 - (-2))^2} = \sqrt{(-18)^2 + (-6 + 2)^2} = \sqrt{324 + 16} = \sqrt{340} = 2\sqrt{85}
\]

#### Side V7
\[
V7 = \sqrt{(-6 - (-8))^2 + (-10 - (-6))^2} = \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

### Step 2: Calculate the slopes of the sides

To find the slopes, we will use the formula:

\[
\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Slope of VA
\[
\text{Slope}_{VA} = \frac{2 - (-6)}{8 - (-8)} = \frac{8}{16} = \frac{1}{2}
\]

#### Slope of AS
\[
\text{Slope}_{AS} = \frac{-2 - 2}{10 - 8} = \frac{-4}{2} = -2
\]

#### Slope of SV
\[
\text{Slope}_{SV} = \frac{-6 - (-2)}{-8 - 10} = \frac{-4}{-18} = \frac{2}{9}
\]

#### Slope of V7
\[
\text{Slope}_{V7} = \frac{-10 - (-6)}{-6 - (-8)} = \frac{-4}{2} = -2
\]

### Step 3: Analyze the results

- The lengths of two sides, AS and V7, are equal: \( 2\sqrt{5} \).
- The slopes of sides AS and V7 are equal: \( -2 \).
- The lengths of sides VA and SV are not equal, so we find that there are two pairs of sides with equal lengths.

### Step 4: Conclusion

Since we found that:
- Two pairs of opposite sides are equal and parallel (indicated by their equal lengths and equal slopes), this proves that the shape is a **Parallelogram**.
- However, we do not have all sides equal or all angles as right angles, hence this cannot be classified further as a Rectangle, Rhombus, or Square.

Thus, the most specific classification of the quadrilateral is:

**Summary: I am a quadrilateral because it has four sides, but more specifically a parallelogram as two pairs of opposite sides are both equal and parallel.**

Answer 3

Sure! Let's go through the classification of the quadrilateral formed by the points \( V(-8, -6) \), \( A(8, 2) \), \( S(10, -2) \), and \( 7(-6, -10) \) step by step, showing all the calculations clearly.

### Step 1: Calculate the lengths of the sides (using the distance formula)

**Distance Formula:**
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### 1. Calculate the distance \( VA \)
\[
VA = \sqrt{(8 - (-8))^2 + (2 - (-6))^2}
\]
\[
= \sqrt{(8 + 8)^2 + (2 + 6)^2} = \sqrt{(16)^2 + (8)^2} = \sqrt{256 + 64} = \sqrt{320} = 8\sqrt{5}
\]

#### 2. Calculate the distance \( AS \)
\[
AS = \sqrt{(10 - 8)^2 + (-2 - 2)^2}
\]
\[
= \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

#### 3. Calculate the distance \( SV \)
\[
SV = \sqrt{(-8 - 10)^2 + (-6 - (-2))^2}
\]
\[
= \sqrt{(-18)^2 + (-6 + 2)^2} = \sqrt{324 + 16} = \sqrt{340} = 2\sqrt{85}
\]

#### 4. Calculate the distance \( V7 \)
\[
V7 = \sqrt{(-6 - (-8))^2 + (-10 - (-6))^2}
\]
\[
= \sqrt{(2)^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

### Summary of Side Lengths
- \( VA = 8\sqrt{5} \)
- \( AS = 2\sqrt{5} \)
- \( SV = 2\sqrt{85} \)
- \( V7 = 2\sqrt{5} \)

### Step 2: Calculate the slopes of the sides (using the slope formula)

**Slope Formula:**
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### 1. Calculate the slope of \( VA \)
\[
\text{Slope}_{VA} = \frac{2 - (-6)}{8 - (-8)} = \frac{2 + 6}{8 + 8} = \frac{8}{16} = \frac{1}{2}
\]

#### 2. Calculate the slope of \( AS \)
\[
\text{Slope}_{AS} = \frac{-2 - 2}{10 - 8} = \frac{-4}{2} = -2
\]

#### 3. Calculate the slope of \( SV \)
\[
\text{Slope}_{SV} = \frac{-6 - (-2)}{-8 - 10} = \frac{-6 + 2}{-18} = \frac{-4}{-18} = \frac{2}{9}
\]

#### 4. Calculate the slope of \( V7 \)
\[
\text{Slope}_{V7} = \frac{-10 - (-6)}{-6 - (-8)} = \frac{-10 + 6}{-6 + 8} = \frac{-4}{2} = -2
\]

### Summary of Slopes
- \( \text{Slope}_{VA} = \frac{1}{2} \)
- \( \text{Slope}_{AS} = -2 \)
- \( \text{Slope}_{SV} = \frac{2}{9} \)
- \( \text{Slope}_{V7} = -2 \)

### Step 3: Analyze the results

#### Check for Parallel Sides
- The slopes \( \text{Slope}_{AS} \) and \( \text{Slope}_{V7} \) are equal at \( -2 \), indicating that sides AS and V7 are parallel.
- Since AS and V7 are of the same length \( 2\sqrt{5} \), they are corresponding sides of the parallelogram.

- \( VA \) and \( SV \) do not have equal slopes and thus are not parallel.

#### Check for Opposite Sides
- Sides AS and V7 are equal and parallel.
- The lengths \( VA = 8\sqrt{5} \) and \( SV = 2\sqrt{85} \) are not equal.

### Step 4: Conclusion

Since we established that:
- There are two pairs of opposite sides that are equal in length and parallel (AS and V7).
- The other two sides do not create equal pairs nor are they equal slopes.

Thus, we conclude that it is a **Parallelogram** as it meets the criteria of having at least one pair of opposite sides that are equal in length and parallel.

**Summary: I am a parallelogram because I have one pair of opposite sides that are parallel and equal in length (AS and V7), making me a specific type of quadrilateral.**