Question

line ( l ) is shown below. 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 ). (b) find the rise, run, and slope given by triangle ( def ). (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. no. they are not equal because the triangles are similar but not congruent. yes. they are equal because the two triangles are similar. yes. they are equal because the two triangles are congruent.

Explanation:

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

• Rise: The vertical change. Count the units from \(C\) to \(B\) (along the \(y -\)axis). If we assume each grid unit is \(1\), from \(y = 6\) to \(y = 18\), rise \(=18 - 6=12\)
• Run: The horizontal change. Count the units from \(C\) to \(A\) (along the \(x -\)axis). From \(x = 2\) to \(x = 6\), run \(=6 - 2 = 4\)
• Slope: Using the formula \(m=\frac{\text{rise}}{\text{run}}\), \(m=\frac{12}{4}=3\)

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

• Rise: The vertical change. Count the units from \(F\) to \(E\) (along the \(y -\)axis). From \(y = 10\) to \(y = 16\), rise \(=16 - 10 = 6\)
• Run: The horizontal change. Count the units from \(F\) to \(D\) (along the \(x -\)axis). From \(x = 4\) to \(x = 6\), run \(=6 - 4=2\)
• Slope: Using the formula \(m = \frac{\text{rise}}{\text{run}}\), \(m=\frac{6}{2}=3\)

Step3: Check the equality of slopes

• The two triangles \(ABC\) and \(DEF\) are similar (by AA similarity, since the corresponding angles are equal as they are formed by the same line \(l\) and the coordinate axes). For similar right - angled triangles used to calculate the slope of a line, \(\frac{\text{rise}_1}{\text{run}_1}=\frac{\text{rise}_2}{\text{run}_2}\)

Answer:

(a) rise: \(12\), run: \(4\), slope: \(3\)
(b) rise: \(6\), run: \(2\), slope: \(3\)
(c) Yes. They are equal because the two triangles are similar.