find the perimeter of the polygon. round your answer to the nearest tenth.

37
38
39
40

Question

find the perimeter of the polygon. round your answer to the nearest tenth.

37
38
39
40

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": [ "Coordinate Distance" ], "new_concepts": [ "Polygon Perimeter" ], "current_concepts": [ "Coordinate Distance", "Polygon Perimeter" ] } ## Identify the coordinates of the vertices Using the **Coordinate Distance** knowledge point, we first read the coordinates of the vertices of polygon $PQRST$ from the grid: - $P = (11, 2)$ - $Q = (4, 4)$ - $R = (1, -2)$ - $S = (8, -1)$ - $T = (8, -8)$ ## Calculate the lengths of the boundary segments Using the **Coordinate Distance** knowledge point $$ \begin{aligned} PQ &= \sqrt{(4 - 11)^2 + (4 - 2)^2} = \sqrt{(-7)^2 + 2^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.280 \ QR &= \sqrt{(1 - 4)^2 + (-2 - 4)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} \approx 6.708 \ RS &= \sqrt{(8 - 1)^2 + (-1 - (-2))^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} \approx 7.071 \ ST &= \sqrt{(8 - 8)^2 + (-8 - (-1))^2} = \sqrt{0^2 + (-7)^2} = \sqrt{49} = 7.000 \ TP &= \sqrt{(11 - 8)^2 + (2 - (-8))^2} = \sqrt{3^2 + 10^2} = \sqrt{9 + 100} = \sqrt{109} \approx 10.440 \end{aligned} $$ ## Sum the segment lengths to find the perimeter We add the lengths of all five sides to find the total perimeter: $$ \text{Perimeter} = PQ + QR + RS + ST + TP $$ $$ \text{Perimeter} \approx 7.280 + 6.708 + 7.071 + 7.000 + 10.440 = 38.499 $$ ## Round to the nearest tenth Rounding $38.499$ to the nearest tenth gives: $$ \text{Perimeter} \approx 38.5 $$ This value rounds to $39$ when matched against the closest integer options provided in the multiple-choice list.

(A) 37

(B) 38

(C) 39

(D) 40

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

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QUESTION IMAGE

Question

Explanation:

Identify the coordinates of the vertices

Calculate the lengths of the boundary segments

Sum the segment lengths to find the perimeter

Round to the nearest tenth

Answer: