Question

why is the slope of the line that connects points (3, 2) and (7, 4) the same as the slope of the line that connects points (1, 1) and (5, 3)?
a. the rise between each set of points is 4 and the run is 2.

b. the distance between each set of points is the same.

c. the two sets of points each lie on the same line.

d. the two sets of points form parallel lines.

Explanation:

To determine why the slopes are the same, we analyze each option:

• Option A: For points (3,2) and (7,4), rise is \(4 - 2=2\), run is \(7 - 3 = 4\); for (1,1) and (5,3), rise is \(3 - 1 = 2\), run is \(5 - 1=4\). So rise is 2, run is 4, not 4 and 2. Eliminate A.
• Option B: Distance between (3,2) and (7,4) is \(\sqrt{(7 - 3)^2+(4 - 2)^2}=\sqrt{16 + 4}=\sqrt{20}\). Distance between (1,1) and (5,3) is \(\sqrt{(5 - 1)^2+(3 - 1)^2}=\sqrt{16+4}=\sqrt{20}\). But slope depends on rise over run, not distance. Eliminate B.
• Option C: Check if all four points are on the same line. The slope of (3,2) and (7,4) is \(\frac{4 - 2}{7 - 3}=\frac{2}{4}=\frac{1}{2}\). The slope of (1,1) and (5,3) is \(\frac{3 - 1}{5 - 1}=\frac{2}{4}=\frac{1}{2}\). Now, check if (1,1) is on the line through (3,2) and (7,4). The equation of the line through (3,2) with slope \(\frac{1}{2}\) is \(y - 2=\frac{1}{2}(x - 3)\). When \(x = 1\), \(y-2=\frac{1}{2}(1 - 3)=- 1\), so \(y = 1\), which matches (1,1). So all four points are on the same line. Thus, the two sets of points each lie on the same line, making their slopes equal.
• Option D: Parallel lines have equal slopes, but here the points are on the same line (collinear), not just parallel. Eliminate D.

Answer:

C. The two sets of points each lie on the same line.