Ask a New Question
Menu

A gridded coordinate plane displays a light-blue shaded triangle, ABC, superimposed on top of a slightly larger light-blue shaded triangle, CDE. The vertices of the smaller triangle are located at approximately (3,2), (3,6), and (9,6). The vertices of the larger triangle are located at approximately (9,8), (9,6), and (12,9). The vertices of both triangles are marked with black dots.  A line, also in black, passes through the points (1, 0) and (12, 9), forming the hypotenuse of both the smaller and larger triangles. The coordinates and vertices of the triangles are clearly labeled.

Triangles ABC and CDE are called Response area triangles.

The slope of the line through AE can be written as Response area

This is a Response area relationship because Response area

The equation of the line AE is written as Response area

1 answer

To solve the problem, let's analyze the information given about triangles ABC and CDE, as well as the line passing through points (1, 0) and (12, 9).

  1. Vertices of Triangle ABC:

    • A(3, 2)
    • B(3, 6)
    • C(9, 6)

  2. Vertices of Triangle CDE:

    • C(9, 6) (common vertex for both triangles)
    • D(9, 8)
    • E(12, 9)

  3. Line passing through (1, 0) and (12, 9):

    • We first find the slope of the line. The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is computed as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In our case: \[ m = \frac{9 - 0}{12 - 1} = \frac{9}{11} \]

Thus, the slope of the line through AE can be written as: Response area: \( \frac{9}{11} \)

  1. Response area relationship: This is a Response area relationship because both triangles ABC and CDE have a common vertex at C(9, 6), and the line AE forms the hypotenuse of both triangles. The same line also passes through points (1, 0) and (12, 9).

  2. Equation of the line AE: Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), with the slope \( m = \frac{9}{11} \) and a point, say (1, 0), we can find the equation of line AE. \[ y - 0 = \frac{9}{11}(x - 1) \] Simplifying this: \[ y = \frac{9}{11}x - \frac{9}{11} \]

Thus, the equation of the line AE is: Response area: \( y = \frac{9}{11}x - \frac{9}{11} \)

To summarize:

  • The slope of the line through AE: \( \frac{9}{11} \)
  • Reason for the Response area relationship: Both triangles share a vertex and their hypotenuse aligns with the line.
  • The equation of the line AE: \( y = \frac{9}{11}x - \frac{9}{11} \)