which characteristics will prove that $\delta def$ is a right, scalene triangle?

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all different lengths, and their slopes are the same.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all different lengths, and the slopes of $\overline{de}$ and $\overline{ef}$ opposite reciprocals.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all the same length, and their slopes are the same.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all the same length, and their slopes are opposite reciprocals.

Question

which characteristics will prove that $\delta def$ is a right, scalene triangle?

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all different lengths, and their slopes are the same.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all different lengths, and the slopes of $\overline{de}$ and $\overline{ef}$ opposite reciprocals.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all the same length, and their slopes are the same.

$\overline{de}$, $\overline{ef}$, and $\overline{df}$ are all the same length, and their slopes are opposite reciprocals.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Coordinate Geometry Proofs", "Right Triangle Properties", "Scalene Triangle Properties" ], "current_concepts": [ "Coordinate Geometry Proofs", "Right Triangle Properties", "Scalene Triangle Properties", "Perpendicular Lines", "Distance Formula" ] } ## Define the conditions for a scalene triangle A triangle is scalene if all three of its sides have different lengths: $$ DE eq EF eq DF $$ ## Define the conditions for a right triangle A triangle is a right triangle if it contains a right angle ($90^\circ$). In coordinate geometry, this is proven if two sides are perpendicular, meaning their slopes are opposite reciprocals: $$ \text{slope}(DE) \cdot \text{slope}(EF) = -1 $$ ## Match with the given options The second option states that $\overline{DE}$, $\overline{EF}$, and $\overline{DF}$ are all different lengths (proving it is scalene), and the slopes of $\overline{DE}$ and $\overline{EF}$ are opposite reciprocals (proving they are perpendicular and form a right angle).

(A) $\overline{DE}$, $\overline{EF}$, and $\overline{DF}$ are all different lengths, and their slopes are the same.

(B) $\overline{DE}$, $\overline{EF}$, and $\overline{DF}$ are all different lengths, and the slopes of $\overline{DE}$ and $\overline{EF}$ are opposite reciprocals.

(C) $\overline{DE}$, $\overline{EF}$, and $\overline{DF}$ are all the same length, and their slopes are the same.

(D) $\overline{DE}$, $\overline{EF}$, and $\overline{DF}$ are all the same length, and their slopes are opposite reciprocals.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Coordinate Geometry Proofs" ] }

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Explanation:

Define the conditions for a scalene triangle

Define the conditions for a right triangle

Match with the given options

Answer: