Math Problem Statement

Line segment AB where A(-2, 2) and B(3, -3)

Solution

To analyze line segment ABAB where A(2,2)A(-2, 2) and B(3,3)B(3, -3), we can calculate the following:

1. Slope of the Line Segment ABAB:

The slope mm of the line segment connecting points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2) is given by:

m=y2y1x2x1=323(2)=55=1m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{3 - (-2)} = \frac{-5}{5} = -1

2. Midpoint of Line Segment ABAB:

The midpoint MM of the line segment is calculated as:

M=(x1+x22,y1+y22)=(2+32,232)=(12,12)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 3}{2}, \frac{2 - 3}{2} \right) = \left( \frac{1}{2}, \frac{-1}{2} \right)

3. Length of Line Segment ABAB:

The length dd of the line segment can be found using the distance formula:

d=(x2x1)2+(y2y1)2=(3(2))2+(32)2=52+(5)2=25+25=50=52d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - (-2))^2 + (-3 - 2)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}

Summary:

  • Slope: 1-1
  • Midpoint: (12,12)\left( \frac{1}{2}, \frac{-1}{2} \right)
  • Length: 525\sqrt{2}

Would you like further details on any of these points?


Here are 5 related questions:

  1. How do you find the equation of the line passing through points AA and BB?
  2. What is the equation of the perpendicular bisector of segment ABAB?
  3. How do you calculate the area of a triangle formed by points AA, BB, and a third point CC?
  4. How can you determine if another point C(x3,y3)C(x_3, y_3) lies on the line segment ABAB?
  5. What are the coordinates of a point PP that divides the segment ABAB in a given ratio?

Tip: When calculating the distance between two points, remember that the distance formula is derived from the Pythagorean theorem applied in the coordinate plane.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Coordinate Geometry

Formulas

Slope formula
Midpoint formula
Distance formula

Theorems

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Suitable Grade Level

Grades 9-12