Question

test review: slope and similar triangles
solve the following problems.
2.
the two right triangles on the grid below are similar.
the two right triangles are similar because both triangles have a slope of...
a \\( \frac{1}{2} \\)
b \\( \frac{2}{1} \\)
c 1
d \\( \frac{1}{1} \\)
3.
which proportion (slope) correctly represents the change in y - values and x - values
down on the graph below?
a \\( -\frac{1}{2} \\)
b \\( \frac{1}{2} \\)
c \\( -\frac{1}{1} \\)
d 2
which of the following best describes the
value of line 1?
\\( \frac{1}{2} \\) because the ratio of the vertical side
length to the horizontal side length for each
triangle is \\( \frac{1}{2} \\)

1. because the ratio of the vertical side

length of triangle lmn to the vertical side
length of triangle pqr is 2
\\( \frac{2}{1} \\) because the ratio of the vertical side
length to the horizontal side length for
both triangles simplifies to \\( \frac{2}{1} \\)
\\( \frac{1}{2} \\), because the ratio of the vertical side
length of triangle lmn to the vertical side
length of triangle pqr is \\( \frac{1}{2} \\)

Explanation:

1. Problem 2 (Slope calculation for similar right - angled triangles):

• Step 1: Recall the slope formula
• The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a right - angled triangle on a coordinate grid, the slope can also be thought of as the ratio of the vertical side (rise) to the horizontal side (run).
• Let's assume for the larger triangle, if we pick two points. Suppose the vertical side (rise) is \(4\) units and the horizontal side (run) is \(2\) units.
• Using the slope formula \(m=\frac{\text{rise}}{\text{run}}\).
• Step 2: Calculate the slope
• Substitute the values of rise and run into the formula. If rise \(y = 4\) and run \(x = 2\), then \(m=\frac{4}{2}=2\).

1. Problem 1 (Slope description):

• Step 1: Recall the slope formula and similar triangles property
• The slope \(m=\frac{\text{vertical side (rise)}}{\text{horizontal side (run)}}\). For similar triangles, the ratios of their corresponding sides are equal.
• In triangle \(LMN\), if the vertical side (rise) is \(2\) and the horizontal side (run) is \(6\), and in triangle \(OPQ\), if the vertical side (rise) is \(1\) and the horizontal side (run) is \(3\).
• The slope \(m=\frac{1}{3}\) (because for similar triangles, the ratio of the vertical side to the horizontal side is the same. \(\frac{2}{6}=\frac{1}{3}\)).

1. Problem 3 (Slope calculation from a graph):

• Step 1: Identify two points on the line
• Let's say two points \((x_1,y_1)=(0, - 2)\) and \((x_2,y_2)=(4, - 1)\).
• Use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• Step 2: Substitute the values into the formula
• Substitute \(x_1 = 0,y_1=-2,x_2 = 4,y_2=-1\) into \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
• \(m=\frac{-1-(-2)}{4 - 0}=\frac{-1 + 2}{4}=\frac{1}{4}\).

Answer:

1. For problem 1: The slope is \(\frac{1}{3}\) (because the ratio of the vertical side to the horizontal side for similar triangles is constant).
2. For problem 2: The slope is \(2\) (using the ratio of vertical to horizontal side of the right - angled triangle).
3. For problem 3: The slope is \(\frac{1}{4}\) (using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with two points on the line).