Question

right triangles abc and def are drawn to measure the slope of the line. 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 similar triangles can have different sizes. no. they are not equal because the triangles are similar but not congruent. yes. they are equal because the two triangles are congruent. yes. they are equal because the two triangles are similar

Explanation:

Step1: Find rise, run and slope for triangle \(ABC\)

• Rise: The vertical change. Point \(B\) has \(y -\)coordinate \(10\) and point \(C\) has \(y -\)coordinate \(12\). So, rise \(=12 - 10=2\)
• Run: The horizontal change. Point \(B\) has \(x -\)coordinate \(2\) and point \(C\) has \(x -\)coordinate \(12\). So, run \(=12 - 2 = 10\)
• Slope: Using the formula \(m=\frac{\text{rise}}{\text{run}}\), \(m=\frac{2}{10}=\frac{1}{5}\)

Step2: Find rise, run and slope for triangle \(DEF\)

• Rise: Point \(E\) has \(y -\)coordinate \(10\) and point \(F\) has \(y -\)coordinate \(11\). So, rise \(=11 - 10 = 1\)
• Run: Point \(E\) has \(x -\)coordinate \(4\) and point \(F\) has \(x -\)coordinate \(9\). So, run \(=9 - 4=5\)
• Slope: Using the formula \(m = \frac{\text{rise}}{\text{run}}\), \(m=\frac{1}{5}\)

Step3: Check if slopes are equal

Since the two right - triangles \(ABC\) and \(DEF\) are similar (corresponding angles are equal as they are formed by the same line). For similar triangles, the ratio of rise to run (slope) is the same.

Answer:

(a) rise: \(2\), run: \(10\), slope: \(\frac{1}{5}\)
(b) rise: \(1\), run: \(5\), slope: \(\frac{1}{5}\)
(c) Yes. They are equal because the two triangles are similar.