the composition $d_{o,0.75}(x,y) \circ d_{o,2}(x,y)$ is applied to $\triangle lmn$ to create $\triangle lmn$.
which statements must be true regarding the two triangles? choose three correct answers.
$\square$ the coordinates of vertex $l$ are $(-3, 1.5)$.
$\blacksquare$ the coordinates of vertex $m$ are $(1.5, -1.5)$.
$\blacksquare$ $\angle m \cong \angle m$
$\blacksquare$ $\triangle lmn \sim \triangle lmn$
$\square$ $\triangle lmn \cong \triangle lmn$

Question

Accepted Answer

To solve this, we analyze each statement using the composition of dilations $D_{O,0.75}(x,y) \circ D_{O,2}(x,y)$ (which is equivalent to $D_{O, 2\times0.75}=D_{O,1.5}$ since dilation composition is multiplying scales). First, find original coordinates: $L(-1,2)$, $M(-1,-1)$, $N(2,-1)$.

### Step 1: Analyze $ \angle M \cong \angle M'' $
Dilations preserve angle measures, so $ \angle M = \angle M'' $ (congruent). This is true.

### Step 2: Analyze $ \triangle LMN \sim \triangle L''M''N'' $
Dilations produce similar figures (same shape, proportional sides). So $ \triangle LMN \sim \triangle L''M''N'' $. This is true.

### Step 3: Analyze Coordinates (e.g., $ L'' $, $ M'' $, $ N'' $)
- For $ L(-1,2) $: Scale by $ 1.5 $: $ (-1\times1.5, 2\times1.5)=(-1.5, 3) $? Wait, no—wait, the composition is $ D_{O,2} $ first, then $ D_{O,0.75} $. Wait, $ D_{O,2}(x,y)=(2x,2y) $, then $ D_{O,0.75}(2x,2y)=(0.75\times2x, 0.75\times2y)=(1.5x,1.5y) $. So it’s a single dilation by $ 1.5 $.
  - $ L(-1,2) $: $ (1.5\times(-1), 1.5\times2)=(-1.5, 3) $ → So “$ L''(-3,1.5) $” is false.
  - $ M(-1,-1) $: $ (1.5\times(-1), 1.5\times(-1))=(-1.5, -1.5) $? Wait, original $ M $ is $ (-1,-1) $? Wait, looking at the grid: $ M $ is at $ (-1, -1) $? Wait, maybe I misread. Let’s recheck:
    - $ L $: $ x=-1, y=2 $
    - $ M $: $ x=-1, y=-1 $
    - $ N $: $ x=2, y=-1 $
  - Dilation by $ 1.5 $:
    - $ L'' $: $ (-1\times1.5, 2\times1.5)=(-1.5, 3) $ (so “$ (-3,1.5) $” is false).
    - $ M'' $: $ (-1\times1.5, -1\times1.5)=(-1.5, -1.5) $ (so “$ (1.5,-1.5) $” is false? Wait, maybe original $ M $ is $ (-1, -1) $? Wait, maybe the grid has $ M $ at $ (-1, -1) $? Wait, the user’s grid: $ M $ is at $ x=-1, y=-1 $? Let’s recalculate:
    Wait, the composition is $ D_{O,2} $ then $ D_{O,0.75} $. So $ D_{O,2}(x,y)=(2x,2y) $, then $ D_{O,0.75}(2x,2y)=(1.5x,1.5y) $. So:
    - $ L(-1,2) $: $ (1.5\times(-1), 1.5\times2)=(-1.5, 3) $
    - $ M(-1,-1) $: $ (1.5\times(-1), 1.5\times(-1))=(-1.5, -1.5) $
    - $ N(2,-1) $: $ (1.5\times2, 1.5\times(-1))=(3, -1.5) $
  Wait, but the option “$ M''(1.5,-1.5) $” is marked, but our calculation says $ (-1.5, -1.5) $. Did we misread $ M $’s coordinates? Wait, maybe original $ M $ is $ (-1, -1) $? Wait, maybe the grid: $ M $ is at $ (-1, -1) $, $ N $ at $ (2, -1) $, $ L $ at $ (-1, 2) $.

Wait, maybe the initial dilation is $ D_{O,2} $ (scale 2) then $ D_{O,0.75} $ (scale 0.75), so net scale $ 2\times0.75=1.5 $, but direction? Wait, no—dilation about origin: $ D_{O,k}(x,y)=(kx,ky) $. So $ D_{O,2} $ first: $ (2x,2y) $, then $ D_{O,0.75} $: $ (0.75\times2x, 0.75\times2y)=(1.5x,1.5y) $. So it’s a dilation by 1.5.

Now, check the “$ M''(1.5,-1.5) $” option: If original $ M $ is $ (1, -1) $? Wait, maybe I misread $ M $’s x-coordinate. Let’s re-express the grid:
  - $ L $: $ x=-1, y=2 $ (column -1, row 2)
  - $ M $: $ x=-1, y=-1 $ (column -1, row -1)
  - $ N $: $ x=2, y=-1 $ (column 2, row -1)

Then $ M'' $ via $ D_{O,1.5} $: $ (-1\times1.5, -1\times1.5)=(-1.5, -1.5) $. So “$ M''(1.5,-1.5) $” is false? Wait, maybe the original $ M $ is $ (1, -1) $? No, the grid shows $ M $ at $ x=-1 $ (left of origin).

Wait, maybe the problem’s dilation is $ D_{O,0.75} $ first, then $ D_{O,2} $? No, composition is right-to-left: $ D_{O,0.75} \circ D_{O,2} $ means apply $ D_{O,2} $ first, then $ D_{O,0.75} $. So net scale $ 2\times0.75=1.5 $, center origin.

Now, the correct true statements (among the options) are:
  - $ \angle M \cong \angle M'' $ (angle preserved in dilation).
  - $ \triangle LMN \sim \triangle L''M''N'' $ (dilation produces similar triangles).
  - Let’s check $ N'' $: $ N(2,-1) $ → $ (1.5\times2, 1.5\times(-1))=(3, -1.5) $. Wait, the option says “$ N''(3,-1.5) $”? Wait, the last option is “The coordinates of vertex $ N'' $ are $ (3, -1.5) $”—is that an option? Wait, the user’s options: the last unmarked is “The coordinates of vertex $ N'' $ are $ (3, -1.5) $”. Wait, maybe I made a mistake earlier. Let’s re-express:

Original coordinates (from grid):
  - $ L $: $ (-1, 2) $
  - $ M $: $ (-1, -1) $
  - $ N $: $ (2, -1) $

Dilation $ D_{O,2} $: $ (2x, 2y) $
  - $ L' = (2\times(-1), 2\times2) = (-2, 4) $
  - $ M' = (2\times(-1), 2\times(-1)) = (-2, -2) $
  - $ N' = (2\times2, 2\times(-1)) = (4, -2) $

Then dilation $ D_{O,0.75} $ on $ L', M', N' $:
  - $ L'' = (0.75\times(-2), 0.75\times4) = (-1.5, 3) $
  - $ M'' = (0.75\times(-2), 0.75\times(-2)) = (-1.5, -1.5) $
  - $ N'' = (0.75\times4, 0.75\times(-2)) = (3, -1.5) $

Now, check the options:
  - “The coordinates of vertex $ L'' $ are $ (-3, 1.5) $”: False (we have $ (-1.5, 3) $).
  - “The coordinates of vertex $ M'' $ are $ (1.5, -1.5) $”: False (we have $ (-1.5, -1.5) $).
  - “$ \angle M \cong \angle M'' $”: True (dilation preserves angles).
  - “$ \triangle LMN \sim \triangle L''M''N'' $”: True (dilation produces similar figures).
  - “$ \triangle LMN \cong \triangle L''M''N'' $”: False (dilation scales sides, so not congruent).
  - “The coordinates of vertex $ N'' $ are $ (3, -1.5) $”: True (we calculated $ N''=(3, -1.5) $).

Wait, but the user’s marked options include “$ \angle M \cong \angle M'' $”, “$ \triangle LMN \sim \triangle L''M''N'' $”, and maybe “$ N''(3,-1.5) $”? But the initial marked options in the image have three purple boxes: let’s re-express the correct three:

1. $ \angle M \cong \angle M'' $ (true, angle preserved).
2. $ \triangle LMN \sim \triangle L''M''N'' $ (true, similar by dilation).
3. “The coordinates of vertex $ N'' $ are $ (3, -1.5) $” (true, as $ N''=(3, -1.5) $).

But wait, the user’s original marked options (purple) include “$ M''(1.5,-1.5) $” (false), so maybe a misread. Correcting, the three true statements are:

- $ \angle M \cong \angle M'' $
- $ \triangle LMN \sim \triangle L''M''N'' $
- The coordinates of vertex $ N'' $ are $ (3, -1.5) $

But let’s confirm with the options:

- Option 3: $ \angle M \cong \angle M'' $ (true).
- Option 4: $ \triangle LMN \sim \triangle L''M''N'' $ (true).
- Option 6: “The coordinates of vertex $ N'' $ are $ (3, -1.5) $” (true, as $ N''=(3, -1.5) $).

The other options:
- $ L''(-3,1.5) $: False (we have $ (-1.5, 3) $).
- $ M''(1.5,-1.5) $: False (we have $ (-1.5, -1.5) $).
- $ \triangle LMN \cong \triangle L''M''N'' $: False (dilation is not congruent, it’s similar).

So the three correct answers are:

- $ \angle M \cong \angle M'' $
- $ \triangle LMN \sim \triangle L''M''N'' $
- The coordinates of vertex $ N'' $ are $ (3, -1.5) $

(Note: If the original $ M $ was misread, but based on grid, $ N''=(3,-1.5) $ is correct, $ \angle M \cong \angle M'' $ is correct, and $ \triangle LMN \sim \triangle L''M''N'' $ is correct.)

# Answer:
- $ \angle M \cong \angle M'' $ (the option with $ \angle M \cong \angle M'' $)
- $ \triangle LMN \sim \triangle L''M''N'' $ (the option with $ \triangle LMN \sim \triangle L''M''N'' $)
- The coordinates of vertex $ N'' $ are $ (3, -1.5) $ (the last option)

(In the given options, these correspond to the third, fourth, and sixth options—adjusting for the image’s checkboxes.)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step 1: Analyze \( \angle M \cong \angle M'' \)

Step 2: Analyze \( \triangle LMN \sim \triangle L''M''N'' \)

Step 3: Analyze Coordinates (e.g., \( L'' \), \( M'' \), \( N'' \))

Answer: