Question
Given the vertices, determine the most specific classification: Quadrilateral, Parallelogram, Rectangle, Rhombus or Square. U(0, 2), N(4, 7), I(6, 3), (3, -2) (1 point) Item 19 Item 20 Item 21 Item 22 Item 23 Quadrilateral Parallelogram Rectangle Rhombus Square
Answers
Answered by
GPT-4o mini
To classify the given quadrilateral with vertices U(0, 2), N(4, 7), I(6, 3), and a fourth vertex which seems to be missing from your data (you mention "(3, -2)" but it's unclear which vertex this represents), I will assume you meant to consider the point (3, -2) as the fourth vertex, which we will call point M.
To classify the quadrilateral, we need to determine the lengths of the sides and the slopes of the sides to check for parallelism, right angles, and equal lengths.
1. **Calculate the lengths of the sides:**
- \( UN \): \( \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \)
- \( NI \): \( \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \)
- \( IM \): \( \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34} \)
- \( MU \): \( \sqrt{(0 - 3)^2 + (2 - (-2))^2} = \sqrt{9 + 16} = \sqrt{25}= 5\)
2. **Calculate the slopes to check for parallelism:**
- Slope of \( UN \): \( \frac{7 - 2}{4 - 0} = \frac{5}{4} \)
- Slope of \( NI \): \( \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2 \)
- Slope of \( IM \): \( \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3} \)
- Slope of \( MU \): \( \frac{2 - (-2)}{0 - 3} = \frac{4}{-3} = -\frac{4}{3} \)
Using the calculated lengths and slopes, we can determine if any sides are parallel (indicating it's a parallelogram) and if any corners form right angles (to check if it's a rectangle).
- For it to be a parallelogram, we need pairs of opposite sides equal in length and/or parallel. Based on the slopes, none of the lines are parallel.
- No sides are also congruent (since all lengths are different).
Thus, we conclude:
- **It is classified as a Quadrilateral**, as it does not meet the specific requirements of being a Parallelogram, Rectangle, Rhombus, or Square.
So the most specific classification of the shape formed by the vertices \( U(0, 2) \), \( N(4, 7) \), \( I(6, 3) \), and \( M(3, -2) \) is **Quadrilateral**.
To classify the quadrilateral, we need to determine the lengths of the sides and the slopes of the sides to check for parallelism, right angles, and equal lengths.
1. **Calculate the lengths of the sides:**
- \( UN \): \( \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \)
- \( NI \): \( \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \)
- \( IM \): \( \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34} \)
- \( MU \): \( \sqrt{(0 - 3)^2 + (2 - (-2))^2} = \sqrt{9 + 16} = \sqrt{25}= 5\)
2. **Calculate the slopes to check for parallelism:**
- Slope of \( UN \): \( \frac{7 - 2}{4 - 0} = \frac{5}{4} \)
- Slope of \( NI \): \( \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2 \)
- Slope of \( IM \): \( \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3} \)
- Slope of \( MU \): \( \frac{2 - (-2)}{0 - 3} = \frac{4}{-3} = -\frac{4}{3} \)
Using the calculated lengths and slopes, we can determine if any sides are parallel (indicating it's a parallelogram) and if any corners form right angles (to check if it's a rectangle).
- For it to be a parallelogram, we need pairs of opposite sides equal in length and/or parallel. Based on the slopes, none of the lines are parallel.
- No sides are also congruent (since all lengths are different).
Thus, we conclude:
- **It is classified as a Quadrilateral**, as it does not meet the specific requirements of being a Parallelogram, Rectangle, Rhombus, or Square.
So the most specific classification of the shape formed by the vertices \( U(0, 2) \), \( N(4, 7) \), \( I(6, 3) \), and \( M(3, -2) \) is **Quadrilateral**.
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