Math Problem Statement
Solution
The problem involves finding the length of segment in a lattice of points, with key points , , , and , where segments and intersect at point . Let's break down how to approach the problem.
Steps:
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Find the coordinates of point where segments and intersect:
The equations of the lines containing the segments and need to be determined first.-
Line :
The slope between and is: Using the point-slope form with point : y - 2 = -\frac{1}{2}(x + 3) \quad \text{(equation of line AB)} Simplifying: -
Line :
The slope between and is: Using the point-slope form with point : y - 1 = 1(x - 1) \quad \text{(equation of line CD)} Simplifying:
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Solve the system of equations: To find the coordinates of point , solve: Substituting into the first equation: Multiply both sides by 2 to eliminate the fraction: Substituting into : Therefore, .
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Find the length of segment : Using the distance formula between points and : Simplifying:
= \sqrt{\left(\frac{1}{3} + \frac{9}{3}\right)^2 + \left(\frac{1}{3} - \frac{6}{3}\right)^2} = \sqrt{\left(\frac{10}{3}\right)^2 + \left(\frac{-5}{3}\right)^2}$$ $$d = \sqrt{\frac{100}{9} + \frac{25}{9}} = \sqrt{\frac{125}{9}} = \frac{\sqrt{125}}{3} = \frac{5\sqrt{5}}{3}$$ Thus, the length of segment $$AE$$ is $$\frac{5\sqrt{5}}{3}$$ units.
Would you like further details or clarification?
Here are 5 related questions to explore:
- How would the solution change if the coordinates of point were different?
- What is the general formula for the intersection of two lines in 2D space?
- How can you apply this approach to finding distances in 3D coordinate systems?
- Can you derive the equation of a line given two points in vector form?
- How would you solve for the length of the diagonal in a 2D lattice grid?
Tip: When solving for the intersection of two lines, always express both lines in a similar form (slope-intercept) for easier substitution.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Distance Formula
Slope of a Line
Formulas
Slope formula: m = (y2 - y1) / (x2 - x1)
Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Line Intersection Theorem
Pythagorean Theorem
Suitable Grade Level
Grades 10-12
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