A triangle has vertices at $L(2,2)$, $M(4,4)$, and $N(1,6)$. The triangle is transformed according to the rule $R_{0, 180^\circ}$.

Which statements are true regarding the transformation? Select three options.

The rule for the transformation is $(x, y) \rightarrow (-x,-y)$.
The coordinates of $L'$ are $(-2,-2)$.
The coordinates of $M'$ are $(-4,4)$.
The coordinates of $N'$ are $(6,-1)$.
The coordinates of $N'$ are $(-1,-6)$.

Question

04-11-2026
Mathematics

Answered

A triangle has vertices at $L(2,2)$, $M(4,4)$, and $N(1,6)$. The triangle is transformed according to the rule $R_{0, 180^\circ}$.

Which statements are true regarding the transformation? Select three options.

The rule for the transformation is $(x, y) \rightarrow (-x,-y)$.
The coordinates of $L'$ are $(-2,-2)$.
The coordinates of $M'$ are $(-4,4)$.
The coordinates of $N'$ are $(6,-1)$.
The coordinates of $N'$ are $(-1,-6)$.

Verified Answer :

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one quarter of the rectangle

GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T00:13:14-05:00

The rotation rule $R_{0, 180^\circ}$ transforms any point $(x, y)$ to $(-x, -y)$.
Applying this rule to vertices $L(2, 2)$, $M(4, 4)$, and $N(1, 6)$ gives the images $L'(-2, -2)$, $M'(-4, -4)$, and $N'(-1, -6)$.
Based on these calculations, the true statements are:

The rule for the transformation is $(x, y) \rightarrow (-x, -y)$.
The coordinates of $L'$ are $(-2, -2)$.
The coordinates of $N'$ are $(-1, -6)$.

Final answer: The first, second, and fifth statements are true.

Explanation

Analyze the transformation rule and initial data.

A transformation $R_{0, 180^\circ}$ represents a rotation of a point about the origin by 180 degrees. The mathematical rule for this rotation is that every point $(x, y)$ is mapped to $(-x, -y)$. We are given the triangle $LMN$ with vertices $L(2, 2)$, $M(4, 4)$, and $N(1, 6)$. We need to determine which of the provided statements are true regarding the images of these vertices under the given transformation.

Evaluate the transformation rule statement.

The rule for the transformation $R_{0, 180^\circ}$ is indeed $(x, y) \rightarrow (-x, -y)$. This is the standard coordinate rule for a 180-degree rotation about the origin. Thus, the first statement is true.

Calculate the images of the vertices.

Now, we apply the rule $(x, y) \rightarrow (-x, -y)$ to each vertex of the triangle:

For vertex $L(2, 2)$: the transformation gives $L' = (-2, -2)$. Comparing this to the options, the statement "The coordinates of $L'$ are $(-2, -2)$" is true.
For vertex $M(4, 4)$: the transformation gives $M' = (-4, -4)$. The statement "The coordinates of $M'$ are $(-4, 4)$" is false.
For vertex $N(1, 6)$: the transformation gives $N' = (-1, -6)$. Comparing this to the options, the statement "The coordinates of $N'$ are $(6, -1)$" is false, and the statement "The coordinates of $N'$ are $(-1, -6)$" is true.

Identify the true statements.

From our evaluation:

The rule $(x, y) \rightarrow (-x, -y)$ is true.
The coordinates of $L'$ are $(-2, -2)$ is true.
The coordinates of $N'$ are $(-1, -6)$ is true.

The three true statements are the first, second, and fifth options provided in the question.

Examples

Rotations are fundamental in computer graphics and animation. For instance, if you are animating a rotating blade or a moving game character, you apply geometric transformations like this one to update the position of the object's vertices on the screen at every frame. By understanding that a $180^\circ$ rotation simply negates both coordinates, you can efficiently calculate the new positions of all points without recomputing complex trigonometry for every frame.