Question
Given the vertices, determine the most specific classification: Quadrilateral, parallelogram, rectangle, rhombus, or square. U(0,2), N(4,7), I(6,3), T(3,-2)
Answers
GPT-4o mini
To classify the shape formed by the vertices U(0, 2), N(4, 7), I(6, 3), and T(3, -2), we will follow these steps:
1. **Calculate the lengths of each side**.
2. **Calculate the slopes of each side to check for parallelism**.
3. **Check for right angles**.
4. **Check for properties of rhombuses and squares**.
### Step 1: Calculate Lengths of Each Side
Let’s calculate the lengths of the segments UN, NI, IT, and TU using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- **Length of UN**:
\[
UN = \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41}
\]
- **Length of NI**:
\[
NI = \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
- **Length of IT**:
\[
IT = \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34}
\]
- **Length of TU**:
\[
TU = \sqrt{(0 - 3)^2 + (2 + 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
### Step 2: Calculate Slopes of Each Side
Next, we calculate the slopes to find out whether opposite sides are parallel.
- **Slope of UN**:
\[
\text{slope}_{UN} = \frac{7 - 2}{4 - 0} = \frac{5}{4}
\]
- **Slope of NI**:
\[
\text{slope}_{NI} = \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2
\]
- **Slope of IT**:
\[
\text{slope}_{IT} = \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3}
\]
- **Slope of TU**:
\[
\text{slope}_{TU} = \frac{2 - (-2)}{0 - 3} = \frac{4}{-3} = -\frac{4}{3}
\]
### Step 3: Check for Parallelism
We check if opposite sides are parallel by comparing their slopes:
- **UN and IT**: Slopes \( \frac{5}{4} \) and \( \frac{5}{3} \) are not equal.
- **NI and TU**: Slopes \( -2 \) and \( -\frac{4}{3} \) are not equal.
Since no pairs of opposite sides are parallel, it is not a parallelogram, rectangle, or rhombus.
### Step 4: Check for Other Properties
Since it is not a parallelogram, we can check if it is a quadrilateral:
1. All four sides have different lengths ( \( \sqrt{41}, 2\sqrt{5}, \sqrt{34}, 5 \)), so it can't be a rhombus or square.
2. The angles are not right angles as no perpendicular slopes (opposite products yielding \(-1\)) are found.
### Conclusion
The shape defined by U, N, I, and T is a **quadrilateral** because it has four sides, but it does not meet the criteria to be classified further as a parallelogram, rectangle, rhombus, or square.
1. **Calculate the lengths of each side**.
2. **Calculate the slopes of each side to check for parallelism**.
3. **Check for right angles**.
4. **Check for properties of rhombuses and squares**.
### Step 1: Calculate Lengths of Each Side
Let’s calculate the lengths of the segments UN, NI, IT, and TU using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- **Length of UN**:
\[
UN = \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41}
\]
- **Length of NI**:
\[
NI = \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]
- **Length of IT**:
\[
IT = \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34}
\]
- **Length of TU**:
\[
TU = \sqrt{(0 - 3)^2 + (2 + 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
### Step 2: Calculate Slopes of Each Side
Next, we calculate the slopes to find out whether opposite sides are parallel.
- **Slope of UN**:
\[
\text{slope}_{UN} = \frac{7 - 2}{4 - 0} = \frac{5}{4}
\]
- **Slope of NI**:
\[
\text{slope}_{NI} = \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2
\]
- **Slope of IT**:
\[
\text{slope}_{IT} = \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3}
\]
- **Slope of TU**:
\[
\text{slope}_{TU} = \frac{2 - (-2)}{0 - 3} = \frac{4}{-3} = -\frac{4}{3}
\]
### Step 3: Check for Parallelism
We check if opposite sides are parallel by comparing their slopes:
- **UN and IT**: Slopes \( \frac{5}{4} \) and \( \frac{5}{3} \) are not equal.
- **NI and TU**: Slopes \( -2 \) and \( -\frac{4}{3} \) are not equal.
Since no pairs of opposite sides are parallel, it is not a parallelogram, rectangle, or rhombus.
### Step 4: Check for Other Properties
Since it is not a parallelogram, we can check if it is a quadrilateral:
1. All four sides have different lengths ( \( \sqrt{41}, 2\sqrt{5}, \sqrt{34}, 5 \)), so it can't be a rhombus or square.
2. The angles are not right angles as no perpendicular slopes (opposite products yielding \(-1\)) are found.
### Conclusion
The shape defined by U, N, I, and T is a **quadrilateral** because it has four sides, but it does not meet the criteria to be classified further as a parallelogram, rectangle, rhombus, or square.