Question

Given the vertices, determine the most specific classification: Quadrilateral, Parallelogram, Rectangle, Rhombus or Square.

Q(−8,1), U(3,5), A(7,−1), Y(−4,−4)


(1 point)
The most specific classification for the above vertices is a
.

Answer 1

To classify the quadrilateral formed by the vertices Q(−8,1), U(3,5), A(7,−1), and Y(−4,−4), we can follow these steps:

1. **Calculate the lengths of the sides** using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. **Calculate the midpoints of the diagonals** to see if they bisect each other.

3. **Check the slopes of the opposite sides** to see if they are parallel.

4. **Check the lengths of diagonals** and if they are equal (for rectangle and square).

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

- **Length of QU**:
\[
d_{QU} = \sqrt{(3 - (-8))^2 + (5 - 1)^2} = \sqrt{(3 + 8)^2 + (5 - 1)^2} = \sqrt{11^2 + 4^2} = \sqrt{121 + 16} = \sqrt{137}
\]

- **Length of UA**:
\[
d_{UA} = \sqrt{(7 - 3)^2 + (-1 - 5)^2} = \sqrt{(4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}
\]

- **Length of AY**:
\[
d_{AY} = \sqrt{(-4 - 7)^2 + (-4 - (-1))^2} = \sqrt{(-11)^2 + (-3)^2} = \sqrt{121 + 9} = \sqrt{130}
\]

- **Length of YQ**:
\[
d_{YQ} = \sqrt{(-8 - (-4))^2 + (1 - (-4))^2} = \sqrt{(-4)^2 + (5)^2} = \sqrt{16 + 25} = \sqrt{41}
\]

### Step 2: Check the midpoints of the diagonals

- **Midpoint of QU**:
\[
M_{QU} = \left( \frac{-8 + 3}{2}, \frac{1 + 5}{2} \right) = \left( \frac{-5}{2}, 3 \right)
\]

- **Midpoint of AY**:
\[
M_{AY} = \left( \frac{7 + (-4)}{2}, \frac{-1 + (-4)}{2} \right) = \left( \frac{3}{2}, -\frac{5}{2} \right)
\]

### Step 3: Check slopes to verify parallelism

- **Slope of QU**:
\[
\text{slope}_{QU} = \frac{5 - 1}{3 - (-8)} = \frac{4}{11}
\]

- **Slope of AY**:
\[
\text{slope}_{AY} = \frac{-4 - (-1)}{-4 - 7} = \frac{-3}{-11} = \frac{3}{11}
\]

- **Slope of UA**:
\[
\text{slope}_{UA} = \frac{-1 - 5}{7 - 3} = \frac{-6}{4} = -\frac{3}{2}
\]

- **Slope of YQ**:
\[
\text{slope}_{YQ} = \frac{1 - (-4)}{-8 - (-4)} = \frac{5}{-4} = -\frac{5}{4}
\]

### Parallel Sides

We have neither pairs of sides parallel nor equal in length.

### Conclusion

The sides have different lengths, and the slopes indicate that no sides are parallel.

Thus, the most specific classification for the shape formed by the vertices Q(−8,1), U(3,5), A(7,−1), and Y(−4,−4) is a **quadrilateral**.