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.
graph of two triangles on a line

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.
graph with a line and a right triangle

1. could these two triangles below lie on the same line?

diagram of two right triangles: first with height 18, base 10; second with height 2, base 3

1. explain whether or not the triangles shown could lie on the same line.

diagram of two right triangles: first with height 72, base 12; second with height 144, base 24

Explanation:

Step1: Analyze Q1 slope property

For triangles on the same line, slope = $\frac{\text{rise}}{\text{run}}$. Similar triangles have proportional rise/run, so slopes are equal. Congruent triangles have identical side lengths, which is not the case here.

Step2: Calculate Q2 slope

Slope = $\frac{\text{change in } y}{\text{change in } x}$. Use points $(4,40)$ and $(7,70)$: $\frac{70-40}{7-4}=\frac{30}{3}=10$. Catherine's $\frac{1}{10}$ is reciprocal, wrong.

Step3: Compare Q3 slopes

Slope of first triangle: $\frac{18}{10}=1.8$. Slope of second triangle: $\frac{2}{3}\approx0.67$. Slopes are not equal, so they can't be on the same line.

Step4: Compare Q4 slopes

Slope of first triangle: $\frac{72}{12}=6$. Slope of second triangle: $\frac{144}{24}=6$. Slopes are equal, so they can be on the same line.

Answer:

1. The slopes of the two triangles are the same.
2. No, she is not correct. The slope is calculated as $\frac{\text{rise}}{\text{run}}$, using points like $(4,40)$ and $(7,70)$ gives $\frac{70-40}{7-4}=10$, not $\frac{1}{10}$.
3. No, the triangles cannot lie on the same line. Their slopes are $\frac{18}{10}=1.8$ and $\frac{2}{3}\approx0.67$, which are not equal.
4. Yes, the triangles could lie on the same line. Their slopes are $\frac{72}{12}=6$ and $\frac{144}{24}=6$, which are equal.