To determine the intersection point of the two lines, we need to find their equations first and solve them simultaneously. From the graph:

Points Identified:

• $A(4, 0)$
• $B(-2, -3)$
• $C(0, 4)$
• $D(2, -4)$

Step 1: Equations of the Lines

1. Line AB: The slope of line $AB$ is: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 0}{-2 - 4} = \frac{-3}{-6} = \frac{1}{2}$ Using point-slope form with point $A(4, 0)$: $y - 0 = \frac{1}{2}(x - 4) \implies y = \frac{1}{2}x - 2$

2. Line CD: The slope of line $CD$ is: $m_{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 4}{2 - 0} = \frac{-8}{2} = -4$ Using point-slope form with point $C(0, 4)$: $y - 4 = -4(x - 0) \implies y = -4x + 4$

Step 2: Solve for Intersection

To find the intersection point, solve the equations: $y = \frac{1}{2}x - 2 \quad \text{and} \quad y = -4x + 4$ Equating the two: $\frac{1}{2}x - 2 = -4x + 4$ Clear fractions by multiplying through by 2: $x - 4 = -8x + 8$ Combine like terms: $x + 8x = 8 + 4 \implies 9x = 12 \implies x = \frac{12}{9} = \frac{4}{3}$ Substitute $x = \frac{4}{3}$ into $y = \frac{1}{2}x - 2$: $y = \frac{1}{2}\left(\frac{4}{3}\right) - 2 = \frac{2}{3} - 2 = \frac{2}{3} - \frac{6}{3} = -\frac{4}{3}$

Step 3: Final Answer

The lines intersect at: $\boxed{\left(\frac{4}{3}, -\frac{4}{3}\right)}$

Would you like further explanation or a graph illustrating the solution?

Related Questions:

1. What is the geometric significance of the intersection point in the quilt design?
2. How would you verify the intersection point algebraically?
3. Can the slopes of the lines tell you if the intersection is unique or parallel lines?
4. How would the calculation change if one of the lines were vertical?
5. How can the intersection point be confirmed graphically?

Tip:

Always simplify fractions early in calculations to avoid errors in algebraic manipulation.