ellie drew $\delta lmn$, in which $m\angle lmn = 90^{\circ}$. she then drew $\delta pqr$, which was a dilation of $\delta lmn$ by a scale factor of $\frac{1}{2}$ from the center of dilation at point $m$. which of these can be used to prove $\delta lmn \sim \delta pqr$ by the aa similarity postulate?

* $m\angle p = m\angle l$; this can be confirmed by translating point $p$ to point $l$.
* $m\angle p = m\angle n$; this can be confirmed by translating point $p$ to point $n$.
* $\overline{lm} = \frac{1}{2}\overline{pq}$; this can be confirmed by translating point $p$ to point $l$.
* $\overline{mn} = \frac{1}{2}\overline{qr}$; this can be confirmed by translating point $r$ to point $n$.

Question

ellie drew $\delta lmn$, in which $m\angle lmn = 90^{\circ}$. she then drew $\delta pqr$, which was a dilation of $\delta lmn$ by a scale factor of $\frac{1}{2}$ from the center of dilation at point $m$. which of these can be used to prove $\delta lmn \sim \delta pqr$ by the aa similarity postulate?

* $m\angle p = m\angle l$; this can be confirmed by translating point $p$ to point $l$.
* $m\angle p = m\angle n$; this can be confirmed by translating point $p$ to point $n$.
* $\overline{lm} = \frac{1}{2}\overline{pq}$; this can be confirmed by translating point $p$ to point $l$.
* $\overline{mn} = \frac{1}{2}\overline{qr}$; this can be confirmed by translating point $r$ to point $n$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Dilation Properties", "AA Similarity Postulate" ], "new_concepts": [], "current_concepts": [ "Dilation Properties", "AA Similarity Postulate", "Geometric Transformations", "Similarity Transformations" ] } ## Analyze the dilation properties Using the *Dilation Properties* knowledge point $$ \begin{aligned} &\triangle PQR \text{ is a dilation of } \triangle LMN \text{ centered at } M \text{ with scale factor } \frac{1}{2}.\ &\text{Dilation preserves angle measures: } m\angle P = m\angle L, \quad m\angle Q = m\angle M = 90^\circ, \quad m\angle R = m\angle N. \end{aligned} $$ ## Apply the AA similarity postulate Using the *AA Similarity Postulate* knowledge point $$ \begin{aligned} &\text{To prove } \triangle LMN \sim \triangle PQR \text{ by AA, we need two pairs of congruent angles.}\ &\text{Pair 1: } \angle M \cong \angle Q \text{ (both are } 90^\circ \text{ since } M \text{ is the center of dilation and } Q \text{ corresponds to } M\text{).}\ &\text{Pair 2: We need either } m\angle P = m\angle L \text{ or } m\angle R = m\angle N. \end{aligned} $$ ## Evaluate the given options Using the *Dilation Properties* and *AA Similarity Postulate* knowledge points $$ \begin{aligned} &\text{Option 1: } m\angle P = m\angle L \text{ can be confirmed by translating } P \text{ to } L \text{ (corresponding vertices).}\ &\text{Option 2: } m\angle P = m\angle N \text{ is incorrect because } P \text{ and } N \text{ are not corresponding vertices.}\ &\text{Options 3 and 4: These involve side length ratios, which are not used for the AA (Angle-Angle) postulate.} \end{aligned} $$

(A) $m\angle P = m\angle L$; this can be confirmed by translating point P to point L.

(B) $m\angle P = m\angle N$; this can be confirmed by translating point P to point N.

(C) $\overline{LM} = \frac{1}{2}\overline{PQ}$; this can be confirmed by translating point P to point L.

(D) $\overline{MN} = \frac{1}{2}\overline{QR}$; this can be confirmed by translating point R to point N.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Similarity Transformations" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the dilation properties

Apply the AA similarity postulate

Evaluate the given options

Answer: