Question

4.3 similar triangles and slope
question 8 of 15 (1 point) | question attempt: 1 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?
yes. they are equal because the two triangles are similar

Explanation:

Part (a) - Triangle \( ABC \)

Step 1: Identify Coordinates

From the graph, let's find the coordinates of points \( A \), \( B \), and \( C \). Assume \( B \) is at \( (10, 3) \), \( A \) is at \( (10, 6) \), and \( C \) is at \( (8, 3) \).

Step 2: Calculate Rise

Rise is the vertical change (change in \( y \)-coordinate). For \( A \) to \( B \), \( \text{Rise} = y_A - y_B = 6 - 3 = 3 \).

Step 3: Calculate Run

Run is the horizontal change (change in \( x \)-coordinate). For \( C \) to \( B \), \( \text{Run} = x_B - x_C = 10 - 8 = 2 \).

Step 4: Calculate Slope

Slope is \( \frac{\text{Rise}}{\text{Run}} \), so \( \text{Slope} = \frac{3}{2} \).

Part (b) - Triangle \( DEF \)

Step 1: Identify Coordinates

Points \( D \), \( E \), and \( F \): \( E \) is at \( (12, 0) \), \( D \) is at \( (12, 9) \), \( F \) is at \( (6, 0) \).

Step 2: Calculate Rise

Rise: \( y_D - y_E = 9 - 0 = 9 \).

Step 3: Calculate Run

Run: \( x_E - x_F = 12 - 6 = 6 \).

Step 4: Calculate Slope

Slope: \( \frac{9}{6} = \frac{3}{2} \).

Part (c)

Step 1: Compare Slopes

From part (a), slope of \( ABC \) is \( \frac{3}{2} \). From part (b), slope of \( DEF \) is \( \frac{3}{2} \). Since the triangles are similar (corresponding angles equal, so their sides are proportional), the ratios (slopes) are equal.

Answer:

(a)
rise: \( 3 \)
run: \( 2 \)
slope: \( \frac{3}{2} \)

(b)
rise: \( 9 \)
run: \( 6 \)
slope: \( \frac{3}{2} \)

(c)
Yes. They are equal because the two triangles are similar (their corresponding sides are proportional, so the ratio of rise to run (slope) is the same).