Question

slope and similar triangles

1. which of the following is a true statement about the triangles shown on the graph?

the slope of the smaller triangle is smaller than the slope of the larger triangle.
the slope of the larger triangle is larger than the slope of the smaller triangle.
the triangles are congruent triangles.
the slopes of the two triangles are the same.

1. catherine used the two points and the right triangle shown to find the slope of the line. catherine found the slope to be \\(\frac{1}{10}\\). is she correct? explain.
2. could these two triangles below lie on the same line?
3. explain whether or not the triangles shown could lie on the same line.

#5 - 10: identify which line from the graph the following right triangles could lie on.

Explanation:

Step1: Q1: Analyze slope property

Triangles on the same line have equal slopes, as slope is $\frac{\text{rise}}{\text{run}}$ and similar triangles have proportional rise/run.

Step2: Q2: Calculate correct slope

Points are $(4,40)$ and $(7,70)$. Slope = $\frac{70-40}{7-4}=\frac{30}{3}=10$.

Step3: Q3: Compare triangle slopes

Slope 1: $\frac{18}{10}=1.8$, Slope 2: $\frac{2}{3}\approx0.67$. Slopes not equal.

Step4: Q4: Compare triangle slopes

Slope 1: $\frac{72}{12}=6$, Slope 2: $\frac{144}{24}=6$. Slopes are equal.

Step5: Q5-10: Find line slopes

Line A: $\frac{9}{2}=4.5$, Line B: $\frac{6}{4}=1.5$, Line C: $\frac{4}{6}=\frac{2}{3}$.

• Q5: Slope = $\frac{120}{90}=\frac{4}{3}$ → No matching line
• Q6: Slope = $\frac{10}{30}=\frac{1}{3}$ → No matching line
• Q7: Slope = $\frac{24}{18}=\frac{4}{3}$ → No matching line
• Q8: Slope = $\frac{14}{21}=\frac{2}{3}$ → Line C
• Q9: Slope = $\frac{24}{36}=\frac{2}{3}$ → Line C
• Q10: Slope = $\frac{16}{48}=\frac{1}{3}$ → No matching line

Answer:

1. The slopes of the two triangles are the same.
2. No. Catherine is incorrect. The correct slope is 10, not $\frac{1}{10}$ (she swapped rise and run).
3. No, the triangles cannot lie on the same line (their slopes are not equal).
4. Yes, the triangles can lie on the same line (their slopes are equal).
5. No matching line
6. No matching line
7. No matching line
8. Line C
9. Line C
10. No matching line