Question

the composition $d_{o,0.75}(x,y) \cdot d_{o,2}(x,y)$ is applied to $\triangle lmn$ to create $\triangle lmn$.
which statements must be true regarding the two triangles? check all that apply.
$\angle m \cong \angle m$
$\triangle lmn \sim \triangle lmn$
$\triangle lmn \cong \triangle lmn$
the coordinates of vertex $l$ are $(-3, 1.5)$.
the coordinates of vertex $n$ are $(3, -1.5)$.
the coordinates of vertex $m$ are $(1.5, -1.5)$.

Explanation:

Step1: Identify original coordinates

From graph: L(-2,2), M(-1,-1), N(2,-1)

Step2: Apply composition of dilations

Dilation composition: scale factor = 0.75×0.2 = 0.15? Wait, no—order: first D₀,₀.₂ then D₀,₀.₇₅? Wait, composition notation: (D₀,₀.₇₅ ∘ D₀,₀.₂)(x,y) = D₀,₀.₇₅(D₀,₀.₂(x,y)) = 0.75×0.2(x,y) = 0.15(x,y)? No, wait 0.75×0.2=0.15? Wait no: 0.750.2=0.15? Wait no, 0.750.2=0.15? Wait let's check each vertex:

• L(-2,2): 0.2(-2,2)=(-0.4,0.4); 0.75(-0.4,0.4)=(-0.3,0.3) → no match with options. Wait maybe typo: 0.75 and 2? 0.75*2=1.5. Let's try scale factor 1.5:
• L(-2,2): 1.5(-2,2)=(-3,3) → no. L(-2,2)→1.5(-2,2)=(-3,3) no. Wait M(-1,-1):1.5(-1,-1)=(-1.5,-1.5) no. N(2,-1):1.5(2,-1)=(3,-1.5) → matches 5th option. L(-2,2):1.5(-2,2)=(-3,3) no. M(-1,-1):1.5(-1,-1)=(-1.5,-1.5) no. Wait dilation order: maybe D₀,₂ then D₀,0.75: 2*0.75=1.5. Yes! So scale factor 1.5.

Check N(2,-1):1.5*(2,-1)=(3,-1.5) → 5th option correct.
Dilation preserves angles and similarity, so ∠M≅∠M'' and △LMN~△L''M''N'' are true.

Step3: Verify true statements

• ∠M≅∠M'': True (dilation preserves angles)
• △LMN~△L''M''N'': True (dilation produces similar figures)
• △LMN≅△L''M''N'': False (dilation changes size)
• L''(-3,1.5): L(-2,2)→1.5*(-2,2)=(-3,3)≠(-3,1.5) → False
• N''(3,-1.5): N(2,-1)→1.5*(2,-1)=(3,-1.5) → True
• M''(1.5,-1.5): M(-1,-1)→1.5*(-1,-1)=(-1.5,-1.5)≠(1.5,-1.5) → False

Answer:

∠M ≅ ∠M'', △LMN ~ △L''M''N'', The coordinates of vertex N'' are (3, -1.5)