Math Problem Statement
The points Nleft parenthesis, 5, comma, 5, right parenthesis(5,5), Oleft parenthesis, 3, comma, minus, 4, right parenthesis(3,−4), and Pleft parenthesis, minus, 2, comma, minus, 1, right parenthesis(−2,−1) form a triangle. Plot the points then click the "Graph Triangle" button., On this question page, you will first graph points on a coordinate grid. Following the graph are two buttons, "Graph Triangle" will drawn lines between your points, and "Reset" will remove all of your points. Once you press Graph Triangle, 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 Find the desired slopes and lengths, then fill in the words that characterize the triangle. Answer Attempt 1 out of 2
slope of NO = slope of OP = slope of PN = length of NO = length of OP = length of PN = Triangle NOP is Submit Answer
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Solution
To solve this problem and characterize the triangle formed by points , , and , we need to find the slopes of the sides , , and , as well as their lengths. Here is the solution step by step:
Step 1: Find the slopes
The slope of a line between two points and is given by:
Slope of :
Slope of :
Slope of :
Step 2: Find the lengths
The distance between two points and is given by:
Length of :
Length of :
Length of :
Step 3: Characterize the triangle
To classify the triangle, compare slopes and lengths:
- Slopes of , , and are not equal, so the triangle is not isosceles.
- Lengths , but , so the triangle is isosceles.
Final Answer:
- Slope of :
- Slope of :
- Slope of :
- Length of :
- Length of :
- Length of :
- Triangle is isosceles.
Would you like a plot of this triangle or further clarification?
Five Related Questions:
- How do you determine if a triangle is right, acute, or obtuse using slopes and lengths?
- How can the midpoint formula be applied to triangle geometry problems?
- What is the significance of perpendicular slopes in characterizing triangles?
- How do you prove that two sides of a triangle are equal using algebra?
- How can you find the area of a triangle using the vertices' coordinates?
Tip: When comparing slopes, remember perpendicular lines have slopes that are negative reciprocals.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Slopes of Lines
Distance Between Two Points
Triangle Classification
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Classification of Triangles by Side Lengths
Slope and Parallel/Perpendicular Line Properties
Suitable Grade Level
Grades 9-11
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