Math Problem Statement

Determine the coordinates of points A and B given the parallel lines g2: y = x + 2 and g3, with additional data from the diagram.

Solution

The problem in the image asks to determine the coordinates of points AA and BB, with the following details:

  • The line g2g_2 has the equation y=x+2y = x + 2.
  • Lines g2g_2 and g3g_3 are parallel.
  • We also observe the line g1g_1, which appears to be the horizontal axis or another given line.

Step-by-step approach:

1. Equation of g2g_2:

The equation of g2g_2 is given as y=x+2y = x + 2, which is a straight line with a slope of 1 and a y-intercept at y=2y = 2.

2. Parallel Lines g2g_2 and g3g_3:

Since g2g_2 and g3g_3 are parallel, they must have the same slope. Therefore, the equation of g3g_3 will also have a slope of 1 but a different y-intercept. Let’s assume the equation of g3g_3 is of the form y=x+cy = x + c, where cc is the unknown y-intercept.

From the diagram, we can estimate that g3g_3 passes through the point (0, 12). Hence, the equation of g3g_3 becomes: g3:y=x+12g_3: y = x + 12

3. Finding Point AA:

The point AA lies on the line g3g_3 and has a y-coordinate of 7 (as given by the diagram). So, we substitute y=7y = 7 into the equation of g3g_3 to find the corresponding x-coordinate of AA:

7=x+127 = x + 12 Solving for xx: x=712=5x = 7 - 12 = -5 Thus, the coordinates of point AA are (5,7)(-5, 7).

4. Finding Point BB:

The point BB lies on the line g2g_2 and has a y-coordinate of 8 (from the diagram). We substitute y=8y = 8 into the equation of g2g_2 to find the corresponding x-coordinate of BB:

8=x+28 = x + 2 Solving for xx: x=82=6x = 8 - 2 = 6 Therefore, the coordinates of point BB are (6,8)(6, 8).

Final Answer:

  • A(5,7)A(-5, 7)
  • B(6,8)B(6, 8)

Would you like more details or have any other questions? Here are 5 related questions to expand your understanding:

  1. What is the significance of parallel lines having the same slope?
  2. How can you find the equation of a line given two points?
  3. How would the solution change if the lines were perpendicular instead of parallel?
  4. Can you derive the slope-intercept form from two points on a line?
  5. 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