Question

The figure shows a line graph and two shaded triangles that are similar:

A line is shown on a coordinate grid. The x axis values are from negative 10 to positive 10 in increments of 2 for each grid line. The y axis values are from negative 5 to positive 5 in increments of 1for each grid line. The line passes through the ordered pairs negative 6, negative 3, and 0, 0, and 6, 3. A shaded right triangle is formed so that its hypotenuse is from ordered pair 0, 0 labeled O to 4, 2 labeled A, one leg is from 0, 0 to 4,0, and the second leg is from 4,0 to 4, 2. Another shaded right triangle is formed with the hypotenuse is from 4, 2 to 6, 3 labeled B, one leg is from 4, 2 to 6, 2, and the second leg is from 6, 2 to 6, 3.
Which statement about the slope of the line is true? (5 points)

It is fraction 1 over 2 throughout the line.
It is 2 throughout the line.
The slope from point O to point A is fraction 1 over 2 time the slope of the line from point A to point B.
The slope from point O to point A is two times the slope of the line from point A to point B.

Answer 1

To determine the slope of the line and the relationships between the slopes mentioned, let's first find the slope of the line passing through the points (-6, -3) and (6, 3).

The slope of a line passing through two points (x1,y1)(x1​,y1​) and (x2,y2)(x2​,y2​) is given by:

Slope=y2−y1x2−x1Slope=x2​−x1​y2​−y1​​

Let's use the points (-6, -3) and (6, 3) to calculate the slope of the line:

Slope=3−(−3)6−(−6)=3+36+6=612=12Slope=6−(−6)3−(−3)​=6+63+3​=126​=21​

So, the slope of the line passing through the points (-6, -3) and (6, 3) is 1221​.

Now, let's analyze the options: It is 1221​ throughout the line.

This statement matches the slope we calculated, so it is true.

It is 2 throughout the line.

This statement is not true since we found the slope to be 1221​ throughout the line.

The slope from point O to point A is 1221​ times the slope of the line from point A to point B.

Let's find the slope from point O to point A. The points are (0, 0) and (4, 2).

Slope (O to A)=2−04−0=24=12Slope (O to A)=4−02−0​=42​=21​

Now, the slope from point A to point B is the slope of the line we calculated, which is also 1221​.

Therefore, the statement is 1221​ times 1221​, which equals 1441​. This statement is not true.

The slope from point O to point A is two times the slope of the line from point A to point B.

As we found earlier, the slope from point O to point A is 1221​ and the slope from point A to point B is also 1221​.

Therefore, the statement is 2×122×21​, which equals 1. This statement is not true.

So, the correct statement about the slope of the line is:

It is 1221​ throughout the line.

Explanation: