The points N, left parenthesis, minus, 9, comma, 3, right parenthesis, comma, O, left parenthesis, minus, 8, comma, minus, 5, right parenthesis, comma, P, left parenthesis, minus, 1, comma, minus, 9, right parenthesisN(−9,3),O(−8,−5),P(−1,−9), and Q, left parenthesis, minus, 2, comma, minus, 1, right parenthesisQ(−2,−1) form quadrilateral NOPQ. Plot the points then click the "Graph Quadrilateral" button., On this question page, you will first graph points on a coordinate grid. Following the graph are two buttons, "Graph Quadrilateral" will drawn lines between your points, and "Reset" will remove all of your points. Once you press Graph Quadrilateral, if your points are correct, the answer area will appear.
Click on the graph to plot a point. Click a point to delete it.
x
y
N
O
P
Q
Find the desired slopes and lengths, then fill in the words that BEST identifies the type of quadrilateral.
Answer
Attempt 1 out of 2

slope of NO = 
length of NO = 
slope of OP = 
length of OP = 
slope of PQ = 
length of PQ = 
slope of QN = 
length of QN = 
Quadrilateral NOPQ can BEST be described as

To determine the type of quadrilateral formed by points \( N(-9, 3), O(-8, -5), P(-1, -9), Q(-2, -1) \), we need to calculate the slopes and lengths of its sides.

### 1. **Slope Calculation:**

The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

- **Slope of NO** (between points \( N(-9, 3) \) and \( O(-8, -5) \)):

\[
\text{slope of NO} = \frac{-5 - 3}{-8 - (-9)} = \frac{-8}{1} = -8
\]

- **Slope of OP** (between points \( O(-8, -5) \) and \( P(-1, -9) \)):

\[
\text{slope of OP} = \frac{-9 - (-5)}{-1 - (-8)} = \frac{-9 + 5}{-1 + 8} = \frac{-4}{7}
\]

- **Slope of PQ** (between points \( P(-1, -9) \) and \( Q(-2, -1) \)):

\[
\text{slope of PQ} = \frac{-1 - (-9)}{-2 - (-1)} = \frac{-1 + 9}{-2 + 1} = \frac{8}{-1} = -8
\]

- **Slope of QN** (between points \( Q(-2, -1) \) and \( N(-9, 3) \)):

\[
\text{slope of QN} = \frac{3 - (-1)}{-9 - (-2)} = \frac{3 + 1}{-9 + 2} = \frac{4}{-7} = -\frac{4}{7}
\]

### 2. **Length Calculation:**

The formula for the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

- **Length of NO**:

\[
\text{length of NO} = \sqrt{(-8 - (-9))^2 + (-5 - 3)^2} = \sqrt{(1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

- **Length of OP**:

\[
\text{length of OP} = \sqrt{(-1 - (-8))^2 + (-9 - (-5))^2} = \sqrt{(7)^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]

- **Length of PQ**:

\[
\text{length of PQ} = \sqrt{(-2 - (-1))^2 + (-1 - (-9))^2} = \sqrt{(-1)^2 + (8)^2} = \sqrt{1 + 64} = \sqrt{65}
\]

- **Length of QN**:

\[
\text{length of QN} = \sqrt{(-9 - (-2))^2 + (3 - (-1))^2} = \sqrt{(-7)^2 + (4)^2} = \sqrt{49 + 16} = \sqrt{65}
\]

### 3. **Conclusion:**

- The slopes of opposite sides \( NO \) and \( PQ \) are both \( -8 \), which means these sides are parallel.
- The slopes of opposite sides \( OP \) and \( QN \) are both \( -\frac{4}{7} \), meaning these sides are also parallel.
- The lengths of all four sides are equal, \( \sqrt{65} \).

Since both pairs of opposite sides are parallel and all sides have equal length, **Quadrilateral NOPQ** is a **rhombus**.

### Summary of Answers:

- **Slope of NO** = \( -8 \)
- **Length of NO** = \( \sqrt{65} \)
- **Slope of OP** = \( -\frac{4}{7} \)
- **Length of OP** = \( \sqrt{65} \)
- **Slope of PQ** = \( -8 \)
- **Length of PQ** = \( \sqrt{65} \)
- **Slope of QN** = \( -\frac{4}{7} \)
- **Length of QN** = \( \sqrt{65} \)
- **Quadrilateral NOPQ** can BEST be described as a **rhombus**.

Would you like further details or clarification on any step?

Here are five related questions:
1. How do you determine if two sides of a quadrilateral are parallel?
2. What is the difference between a rhombus and a square?
3. Can a rhombus have right angles? If so, under what conditions?
4. What is the general formula for the area of a rhombus?
5. How does the Pythagorean theorem apply to calculating distances in the coordinate plane?

**Tip:** Remember, in any quadrilateral, the slopes of parallel sides will always be equal!

The points N(-9,3), O(-8,-5), P(-1,-9), and Q(-2,-1) form quadrilateral NOPQ. Plot the points then click the 'Graph Quadrilateral' button. Find the desired slopes and lengths, then fill in the words that BEST identifies the type of quadrilateral.

Learn how to identify the type of quadrilateral formed by the points N(-9,3), O(-8,-5), P(-1,-9), and Q(-2,-1) by calculating slopes and lengths. Understand key concepts of coordinate geometry including slope and distance formulas, suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation