Question

4.3 similar triangles and slope
question 9 of 15 (1 point) | question attempt: 2 of 5
complete the parts below.
(a) find the rise, run, and slope given by triangle abc.
rise:
run:
slope:
(b) find the rise, run, and slope given by triangle def.
rise:
run:
slope:
(c) are the two slopes computed above equal? why or why not?
no. they are not equal because the larger the triangle, the larger the slope.

Explanation:

Step1: Analyze Triangle ABC

First, identify coordinates of points. Let's assume \( C(0, 2) \), \( A(4, 2) \), \( B(4, 10) \).
Rise: Vertical change from \( A \) to \( B \): \( 10 - 2 = 8 \).
Run: Horizontal change from \( C \) to \( A \): \( 4 - 0 = 4 \).
Slope: \( \frac{\text{rise}}{\text{run}} = \frac{8}{4} = 2 \).

Step2: Analyze Triangle DEF

Points: Let \( F(1, 4) \), \( D(2, 4) \), \( E(2, 6) \).
Rise: Vertical change from \( D \) to \( E \): \( 6 - 4 = 2 \).
Run: Horizontal change from \( F \) to \( D \): \( 2 - 1 = 1 \).
Slope: \( \frac{\text{rise}}{\text{run}} = \frac{2}{1} = 2 \).

Step3: Compare Slopes

Since both slopes are \( 2 \), they are equal. Similar triangles have proportional sides, so slope (rise/run) remains constant.

Part (a)

rise: \( 8 \), run: \( 4 \), slope: \( 2 \)

Part (b)

rise: \( 2 \), run: \( 1 \), slope: \( 2 \)

Part (c)

Yes, the slopes are equal. Because triangles \( ABC \) and \( DEF \) are similar, their corresponding sides are proportional, so \( \frac{\text{rise}}{\text{run}} \) (slope) is the same for both.

Answer:

s:
(a) rise: \( \boldsymbol{8} \), run: \( \boldsymbol{4} \), slope: \( \boldsymbol{2} \)
(b) rise: \( \boldsymbol{2} \), run: \( \boldsymbol{1} \), slope: \( \boldsymbol{2} \)
(c) Yes, because the triangles are similar (proportional sides, so slope = rise/run is equal).