Question

Points and are images of points A and B after a 270 rotation about the origin. If the slope of is ​3, what is the slope of ​? Explain.

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Part 1
Select the correct choice below and fill in the answer box to complete your choice.
A.
A rotation of 270 would result in a line parallel to . Since the slopes of parallel lines are​ equal, the slope of is

7.
B.
A rotation of 270 would result in a line parallel to . Since the slopes of parallel lines are opposite​ reciprocals, the slope of is

enter your response here.
C.
A rotation of 270 would result in a line perpendicular to . Since the slopes of perpendicular lines are opposite​ reciprocals, the slope of is

enter your response here.
D.
A rotation of 270 would result in a line perpendicular to . Since the slopes of perpendicular lines are​ equal, the slope of is

enter your response here.

Answer 1

The slope of a line can be determined by calculating the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. To find the slope of line , we can examine the relationship between points A and B.

Given that point is the image of point A after a 270° rotation about the origin, we can conclude that points A and are related by a 90° rotation about the origin. This is because a 270° rotation is equivalent to a 90° rotation in the opposite direction.

When a point is rotated 90° counterclockwise or clockwise about the origin, the new coordinates can be found by swapping the x and y coordinates and changing the sign of one of them.

Let's assume the coordinates of point A are (x1, y1) and the coordinates of point are (x2, y2).

To find the coordinates of point , we can use the rotation rule:

x2 = -y1

y2 = x1

Now, let's substitute the given slope of line (m) into the slope formula and calculate the slope of line .

The slope of a line can be represented by the formula:

m = (y2 - y1) / (x2 - x1)

Since the coordinates of point A are (x1, y1) and the coordinates of point are (x2, y2), we can rewrite the slope formula as:

m = (y - y1) / (x - x1)

Now, let's substitute the values we obtained for the coordinates of points A and into the slope formula:

m = (y2 - y1) / (x2 - x1)

m = (x1 - (-y1)) / (y1 - x1)

Simplifying the expression, we get:

m = (x1 + y1) / (y1 - x1)

Therefore, the slope of line is (x1 + y1) / (y1 - x1).

Explanation: