Question

line ( l ) is shown below.
right triangles ( a b c ) and ( d e f ) are drawn to measure the slope of the line.

complete the parts below.
(a) find the rise, run, and slope given by triangle ( a b c ).
rise: ( square ) run: ( square ) slope: ( square )
(b) find the rise, run, and slope given by triangle ( d e f ).
rise: ( square ) run: ( square ) slope: ( square )
(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. From \(A\) to \(B\), \(y\) - values change from \(10\) to \(16\). So, rise \(=16 - 10=6\)
• Run: The horizontal change. From \(A\) to \(B\), \(x\) - values change from \(2\) to \(3\). So, run \(=3 - 2 = 1\)
• Slope: Using the formula \(m=\frac{\text{rise}}{\text{run}}\), \(m=\frac{6}{1}=6\)

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

• Rise: The vertical change. From \(D\) to \(E\), \(y\) - values change from \(12\) to \(14\). So, rise \(=14 - 12 = 2\)
• Run: The horizontal change. From \(D\) to \(E\), \(x\) - values change from \(1\) to \(\frac{4}{3}\). So, run \(=\frac{4}{3}-1=\frac{1}{3}\)
• Slope: Using the formula \(m = \frac{\text{rise}}{\text{run}}\), \(m=\frac{2}{\frac{1}{3}}=6\)

Step3: Analyze the slopes

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

Answer:

(a) rise: \(6\), run: \(1\), slope: \(6\)
(b) rise: \(2\), run: \(\frac{1}{3}\), slope: \(6\)
(c) Yes. They are equal because the two triangles are similar.