planes x and y are perpendicular. points a, e, f, and g are points only in plane x. points r and s are points in both planes x and y. lines ea and fg are parallel.

based on this information, which pair of lines, together, could be perpendicular to $\overleftrightarrow{rs}$? select two options.

$\square\ \overleftrightarrow{ea}$
$\square\ \overleftrightarrow{fg}$
$\square\ \overleftrightarrow{ef}$
$\square\ \overleftrightarrow{ag}$
$\square\ \overleftrightarrow{rs}$

Question

planes x and y are perpendicular. points a, e, f, and g are points only in plane x. points r and s are points in both planes x and y. lines ea and fg are parallel.

based on this information, which pair of lines, together, could be perpendicular to $\overleftrightarrow{rs}$? select two options.

$\square\ \overleftrightarrow{ea}$
$\square\ \overleftrightarrow{fg}$
$\square\ \overleftrightarrow{ef}$
$\square\ \overleftrightarrow{ag}$
$\square\ \overleftrightarrow{rs}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "3D Geometry", "Parallel Planes and Lines" ], "new_concepts": [ "Line perpendicular to a plane", "Perpendicular planes properties" ], "current_concepts": [ "3D Geometry", "Parallel Planes and Lines", "Line perpendicular to a plane", "Perpendicular planes properties" ] } ## Analyze the geometric configuration $$ \begin{aligned} &\text{Plane } X \perp \text{Plane } Y\ &\overleftrightarrow{RS} = X \cap Y \quad (\text{intersection line})\ &A, E, F, G \in X \text{ only}\ &\overleftrightarrow{EA} \parallel \overleftrightarrow{FG} \end{aligned} $$ ## Determine perpendicularity to the intersection line $$ \begin{aligned} &\text{Since } X \perp Y \text{ and } \overleftrightarrow{RS} = X \cap Y:\ &\text{Any line in plane } X \text{ that is perpendicular to the intersection line } \overleftrightarrow{RS} \text{ is perpendicular to plane } Y.\ &\text{From the diagram, the vertical lines } \overleftrightarrow{EA} \text{ and } \overleftrightarrow{FG} \text{ in plane } X \text{ are perpendicular to the horizontal intersection line } \overleftrightarrow{RS}.\ &\overleftrightarrow{EA} \perp \overleftrightarrow{RS} \quad \text{and} \quad \overleftrightarrow{FG} \perp \overleftrightarrow{RS} \end{aligned} $$ ## Identify the correct options $$ \begin{aligned} &\text{The options shown are: } \overleftrightarrow{EA}, \overleftrightarrow{ER}, \overleftrightarrow{EF}, \overleftrightarrow{FG}, \overleftrightarrow{FS}\ &\text{Therefore, the two lines perpendicular to } \overleftrightarrow{RS} \text{ are } \overleftrightarrow{EA} \text{ and } \overleftrightarrow{FG}. \end{aligned} $$

(A) $\overleftrightarrow{EA}$

(B) $\overleftrightarrow{ER}$

(C) $\overleftrightarrow{EF}$

(D) $\overleftrightarrow{FG}$

(E) $\overleftrightarrow{FS}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "3D Geometry" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the geometric configuration

Determine perpendicularity to the intersection line

Identify the correct options

Answer: