which points are a distance of 5 units from $(2, 3)$ if $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$?

$(7, 3)$
$(-1, -1)$
$(0, 3)$
$(2, 8)$
$(-7, 6)$

Question

which points are a distance of 5 units from $(2, 3)$ if $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$?

$(7, 3)$
  $(-1, -1)$
  $(0, 3)$
  $(2, 8)$
  $(-7, 6)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Distance Formula", "Coordinate Geometry" ], "current_concepts": [ "Distance Formula", "Coordinate Geometry" ] } ## State the distance formula and given point $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ $$ (x_1, y_1) = (2, 3),\quad d = 5 $$ $$ (x_2 - 2)^2 + (y_2 - 3)^2 = 25 $$ ## Test each given coordinate option $$ \begin{aligned} &\text{For } (7, 3): (7-2)^2 + (3-3)^2 = 5^2 + 0^2 = 25 \quad (\text{Valid})\ &\text{For } (-1, -1): (-1-2)^2 + (-1-3)^2 = (-3)^2 + (-4)^2 = 9 + 16 = 25 \quad (\text{Valid})\ &\text{For } (0, 3): (0-2)^2 + (3-3)^2 = (-2)^2 + 0^2 = 4 e 25 \quad (\text{Invalid})\ &\text{For } (2, 8): (2-2)^2 + (8-3)^2 = 0^2 + 5^2 = 25 \quad (\text{Valid})\ &\text{For } (-7, 6): (-7-2)^2 + (6-3)^2 = (-9)^2 + 3^2 = 81 + 9 = 90 e 25 \quad (\text{Invalid}) \end{aligned} $$ ## Identify the correct points $$ (7, 3), \quad (-1, -1), \quad (2, 8) $$

(A) (7, 3)

(B) (-1, -1)

(C) (0, 3)

(D) (2, 8)

(E) (-7, 6)

{ "elements": [ { "type": "point", "params": [[2, 3]], "properties": { "name": "C(2,3)", "size": 5, "color": "#8C55F2", "withLabel": true } }, { "type": "circle", "params": [[2, 3], 5], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "dash": 2 } }, { "type": "point", "params": [[7, 3]], "properties": { "name": "(7,3)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[-1, -1]], "properties": { "name": "(-1,-1)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[2, 8]], "properties": { "name": "(2,8)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [[0, 3]], "properties": { "name": "(0,3)", "size": 3, "color": "#777777", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Distance Formula" ] }

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QUESTION IMAGE

Question

Explanation:

State the distance formula and given point

Test each given coordinate option

Identify the correct points

Answer: