Math / Geometry

QuestionMathXL for School: Practice and Problem-Solving
Points $A^{\prime}$ and $B^{\prime}$ are images of points $A$ and $B$ after a $270^{\circ}$ rotation about the origin. If the slope of $\overrightarrow{A B}$ is -3 , what is the slope of $\widehat{A^{\prime} B^{\prime}}$ ? Explain.
Select the correct choice below and fill in the answer box to complete your chaice. A. A rotation of $270^{\circ}$ would result in a line parallel to $\overrightarrow{A B}$ . Since the slopes of parallel lines are equal, the slope of $\overrightarrow{\mathrm{A}^{\prime} B^{\prime}}$ is $\square$ 1. B. A rotation of $270^{\circ}$ would result in a line perpendicular to $\overrightarrow{\mathrm{AB}}$ . Since the slopes of perpendicular lines are opposite reciprocals, the slope of $\overrightarrow{\mathrm{A}^{\prime} \mathrm{B}^{\prime}}$ is $\square$ . C. A rotation of $270^{\circ}$ would result in a line perpendicular to $\overrightarrow{A B}$ . Since the slopes of perpendicular lines are equal, the slope of $\overrightarrow{A^{\prime} B}$ is $\square$ . D. A rotation of $270^{\circ}$ would result in a line parallel to $\overrightarrow{A B}$ . Since the slopes of parallel lines are opposite reciprocals, the slope of $\overrightarrow{A^{\prime} B}$ is $\square$

Studdy Solution

STEP 1

1. The rotation is about the origin.
2. The rotation angle is $270^\circ$ .
3. The slope of $\overrightarrow{AB}$ is $-3$ .

STEP 2

1. Understand the effect of a $270^\circ$ rotation on the slope.
2. Determine the relationship between the slopes of the original and rotated lines.
3. Choose the correct answer based on the relationship.

STEP 3

Understand the effect of a $270^\circ$ rotation on the slope:
A $270^\circ$ rotation about the origin is equivalent to a $90^\circ$ counterclockwise rotation. A $90^\circ$ rotation changes the slope of a line to the negative reciprocal of the original slope.

STEP 4

Determine the relationship between the slopes of the original and rotated lines:
Given the slope of $\overrightarrow{AB}$ is $-3$ , the slope of a line perpendicular to it (after a $90^\circ$ rotation) is the negative reciprocal. The negative reciprocal of $-3$ is $\frac{1}{3}$ .

STEP 5

Choose the correct answer based on the relationship:
Since a $270^\circ$ rotation results in a line perpendicular to the original, and the slopes of perpendicular lines are opposite reciprocals, the correct choice is:
B. A rotation of $270^\circ$ would result in a line perpendicular to $\overrightarrow{\mathrm{AB}}$ . Since the slopes of perpendicular lines are opposite reciprocals, the slope of $\overrightarrow{\mathrm{A}^{\prime} \mathrm{B}^{\prime}}$ is $\frac{1}{3}$ .
The slope of $\overrightarrow{A^{\prime}B^{\prime}}$ is:
$\boxed{\frac{1}{3}}$

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