To solve this problem, we'll determine the slopes and lengths of the sides of the triangle formed by the points $N(5, 5)$, $O(3, -4)$, and $P(-2, -1)$. Then, we'll characterize the triangle based on its properties.

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1. Slope Calculations

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.$

a) Slope of $NO$:

$\text{slope of } NO = \frac{-4 - 5}{3 - 5} = \frac{-9}{-2} = 4.5.$

b) Slope of $OP$:

$\text{slope of } OP = \frac{-1 - (-4)}{-2 - 3} = \frac{3}{-5} = -0.6.$

c) Slope of $PN$:

$\text{slope of } PN = \frac{5 - (-1)}{5 - (-2)} = \frac{6}{7} \approx 0.857.$

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2. Length Calculations

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$

a) Length of $NO$:

$\text{length of } NO = \sqrt{(3 - 5)^2 + (-4 - 5)^2} = \sqrt{(-2)^2 + (-9)^2} = \sqrt{4 + 81} = \sqrt{85}.$

b) Length of $OP$:

$\text{length of } OP = \sqrt{(-2 - 3)^2 + (-1 - (-4))^2} = \sqrt{(-5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34}.$

c) Length of $PN$:

$\text{length of } PN = \sqrt{(5 - (-2))^2 + (5 - (-1))^2} = \sqrt{(7)^2 + (6)^2} = \sqrt{49 + 36} = \sqrt{85}.$

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3. Characterizing the Triangle

To characterize the triangle, compare slopes and side lengths:

1. The slopes of $NO$, $OP$, and $PN$ are distinct, so the triangle is not right-angled.
2. Two side lengths ($\sqrt{85}$) are equal, and one side length ($\sqrt{34}$) is different. Thus, the triangle is isosceles.

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Final Answers:

• Slope of $NO$: $4.5$
• Slope of $OP$: $-0.6$
• Slope of $PN$: $0.857$
• Length of $NO$: $\sqrt{85}$
• Length of $OP$: $\sqrt{34}$
• Length of $PN$: $\sqrt{85}$
• Triangle $NOP$ is: Isosceles

Would you like more clarification or need assistance plotting the points on the graph?

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Related Questions:

1. How do you determine if a triangle is right-angled using slopes?
2. What is the significance of calculating the slopes for triangle sides?
3. How can you verify the isosceles property using side lengths?
4. What are other methods to characterize a triangle using its vertices?
5. Can you calculate the area of the triangle given these points?

Tip:

When finding slopes or lengths, always double-check signs for accuracy, especially when dealing with negative coordinates!