Math Problem Statement
Solution
The problem in the image asks to determine the coordinates of points and , with the following details:
- The line has the equation .
- Lines and are parallel.
- We also observe the line , which appears to be the horizontal axis or another given line.
Step-by-step approach:
1. Equation of :
The equation of is given as , which is a straight line with a slope of 1 and a y-intercept at .
2. Parallel Lines and :
Since and are parallel, they must have the same slope. Therefore, the equation of will also have a slope of 1 but a different y-intercept. Let’s assume the equation of is of the form , where is the unknown y-intercept.
From the diagram, we can estimate that passes through the point (0, 12). Hence, the equation of becomes:
3. Finding Point :
The point lies on the line and has a y-coordinate of 7 (as given by the diagram). So, we substitute into the equation of to find the corresponding x-coordinate of :
Solving for : Thus, the coordinates of point are .
4. Finding Point :
The point lies on the line and has a y-coordinate of 8 (from the diagram). We substitute into the equation of to find the corresponding x-coordinate of :
Solving for : Therefore, the coordinates of point are .
Final Answer:
Would you like more details or have any other questions? Here are 5 related questions to expand your understanding:
- What is the significance of parallel lines having the same slope?
- How can you find the equation of a line given two points?
- How would the solution change if the lines were perpendicular instead of parallel?
- Can you derive the slope-intercept form from two points on a line?
- How can you determine the distance between two points on a graph?
Tip: When solving geometry problems involving parallel lines, always check if the slopes are equal to ensure the lines don't intersect.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Linear Equations
Parallel Lines
Formulas
Slope-intercept form of a line: y = mx + c
Equation of parallel lines: same slope (m)
Theorems
Properties of Parallel Lines
Suitable Grade Level
Grade 8-10
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