Question

1. △lmn is shown in the coordinate plane.△lmn is the image of △lmn under a reflection across the y - axis followed by a dilat centered at the origin with scale factor 2.part agrap △lmn in the coordinate plane.part b△lmn is ____ to △lmn

Explanation:

Part A (Graphing $\boldsymbol{\triangle L''M''N''}$)

To graph $\triangle L''M''N''$, we first need the coordinates of $L$, $M$, and $N$ from $\triangle LMN$. Let's assume the coordinates (from the graph):

• Let $L = (-6, 1)$, $M = (-5, 4)$, $N = (-2, 3)$ (estimating from the grid).

Step 1: Reflect across the $y$-axis

The reflection of a point $(x, y)$ across the $y$-axis is $(-x, y)$.

• $L'$ (after reflection): $(-(-6), 1) = (6, 1)$
• $M'$ (after reflection): $(-(-5), 4) = (5, 4)$
• $N'$ (after reflection): $(-(-2), 3) = (2, 3)$

Step 2: Dilate with scale factor 2 (center at origin)

Dilation of a point $(x, y)$ with scale factor $k$ is $(kx, ky)$.

• $L''$: $(2 \times 6, 2 \times 1) = (12, 2)$
• $M''$: $(2 \times 5, 2 \times 4) = (10, 8)$
• $N''$: $(2 \times 2, 2 \times 3) = (4, 6)$

Plot these points $(12, 2)$, $(10, 8)$, and $(4, 6)$ and connect them to form $\triangle L''M''N''$.

Part B (Determining Similarity/Congruence)

• Reflection is a rigid transformation (preserves shape and size, so congruent).
• Dilation with scale factor 2 changes the size (scales lengths by 2) but preserves shape (angles remain equal, sides are proportional).

Congruent figures have equal size and shape; similar figures have proportional sides and equal angles. Since dilation scales the figure, $\triangle L''M''N''$ is not congruent to $\triangle LMN$ (different sizes) but is similar (same shape, proportional sides).

Final Answers

• Part A: Graph $\triangle L''M''N''$ with vertices at $(12, 2)$, $(10, 8)$, $(4, 6)$.
• Part B: $\triangle L''M''N''$ is $\boldsymbol{\text{similar}}$ to $\triangle LMN$.

(For Part B, circle "similar" in the box.)

Answer:

Part A (Graphing $\boldsymbol{\triangle L''M''N''}$)

To graph $\triangle L''M''N''$, we first need the coordinates of $L$, $M$, and $N$ from $\triangle LMN$. Let's assume the coordinates (from the graph):

• Let $L = (-6, 1)$, $M = (-5, 4)$, $N = (-2, 3)$ (estimating from the grid).

Step 1: Reflect across the $y$-axis

The reflection of a point $(x, y)$ across the $y$-axis is $(-x, y)$.

• $L'$ (after reflection): $(-(-6), 1) = (6, 1)$
• $M'$ (after reflection): $(-(-5), 4) = (5, 4)$
• $N'$ (after reflection): $(-(-2), 3) = (2, 3)$

Step 2: Dilate with scale factor 2 (center at origin)

Dilation of a point $(x, y)$ with scale factor $k$ is $(kx, ky)$.

• $L''$: $(2 \times 6, 2 \times 1) = (12, 2)$
• $M''$: $(2 \times 5, 2 \times 4) = (10, 8)$
• $N''$: $(2 \times 2, 2 \times 3) = (4, 6)$

Plot these points $(12, 2)$, $(10, 8)$, and $(4, 6)$ and connect them to form $\triangle L''M''N''$.

Part B (Determining Similarity/Congruence)

• Reflection is a rigid transformation (preserves shape and size, so congruent).
• Dilation with scale factor 2 changes the size (scales lengths by 2) but preserves shape (angles remain equal, sides are proportional).

Congruent figures have equal size and shape; similar figures have proportional sides and equal angles. Since dilation scales the figure, $\triangle L''M''N''$ is not congruent to $\triangle LMN$ (different sizes) but is similar (same shape, proportional sides).

Final Answers

• Part A: Graph $\triangle L''M''N''$ with vertices at $(12, 2)$, $(10, 8)$, $(4, 6)$.
• Part B: $\triangle L''M''N''$ is $\boldsymbol{\text{similar}}$ to $\triangle LMN$.

(For Part B, circle "similar" in the box.)