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Line

l
is shown below.Right triangles
ABC
and
DEF
are drawn to measure the slope of the line.

y
2
4
6
8
10
12
14
16
18
20
22
-2
x
2
4
6
8
10
12
14
16
18
20
22
-2
B
A
C
E
D
F
l
Complete the parts below.

(a) Find the run, rise, and slope given by triangle
ABC
.
run:
rise:
slope:
(b) Find the run, rise, and slope given by triangle
DEF
.
run:
rise:
slope:
(c) Are the two slopes computed above equal? Why or why not?
Yes. They are equal because the two triangles are similar.
Yes. They are equal because the two triangles are congruent.
No. They are not equal because similar triangles can have different sizes.
No. They are not equal because the larger the triangle, the smaller the slope.

1 answer

To answer the questions, I'll need to determine the run, rise, and slope of the given triangles \( ABC \) and \( DEF \) based on their coordinates. Since I do not have the actual coordinates, I'll guide you through the typical process instead.

Let’s define the terms:

  1. Run: The horizontal distance between two points.
  2. Rise: The vertical distance between two points.
  3. Slope: The ratio of the rise to the run, usually calculated as \( \text{slope} = \frac{\text{rise}}{\text{run}} \).

(a) Triangle \( ABC \)

Assuming you have the coordinates of points \( A \) and \( C \):

  • Run: \( \text{Run} = x_C - x_A \)
  • Rise: \( \text{Rise} = y_C - y_A \)
  • Slope: \( \text{Slope} = \frac{\text{Rise}}{\text{Run}} \)

(b) Triangle \( DEF \)

Assuming you have the coordinates of points \( D \) and \( F \):

  • Run: \( \text{Run} = x_F - x_D \)
  • Rise: \( \text{Rise} = y_F - y_D \)
  • Slope: \( \text{Slope} = \frac{\text{Rise}}{\text{Run}} \)

Please provide the specific coordinates for the points \( A, B, C, D, E, F \) or any specific measurements from the graph so that I can help you find the run, rise, and slope values for both triangles.

(c) Comparing slopes

To answer the question regarding the equality of the slopes of triangles \( ABC \) and \( DEF \):

  • If both triangles \( ABC \) and \( DEF \) are drawn from the same line, then the slopes will be equivalent regardless of the triangle's size since slope is a direction and not dependent on the distance or size of the triangles.

Thus, the correct statement could be:

  • Yes. They are equal because the two triangles are similar.

This implies that they are both measuring the same incline of the line \( l \) despite their different sizes.