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On a coordinate grid, point P has coordinates (3,2). Points Q and R have coordinates (8,6) and (4,-5) respectively. Which of points Q and R is closer to point P? You must show full working to support your answer.

Sagot :

Answer: Point Q is closer

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Explanation:

Use the distance formula to calculate the length of segment PQ

[tex]P = (x_1,y_1) = (3,2) \text{ and }Q = (x_2,y_2) = (8,6)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(3-8)^2 + (2-6)^2}\\\\d = \sqrt{(-5)^2 + (-4)^2}\\\\d = \sqrt{25 + 16}\\\\d = \sqrt{41}\\\\d \approx 6.40312\\\\[/tex]

PQ is roughly 6.403 units long.

Repeat the same type of calculations, but this time we want to find the length of segment PR.

[tex]P = (x_1,y_1) = (3,2) \text{ and }R = (x_2,y_2) = (4,-5)\\\\d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(3-4)^2 + (2-(-5))^2}\\\\d = \sqrt{(3-4)^2 + (2+5)^2}\\\\d = \sqrt{(-1)^2 + (7)^2}\\\\d = \sqrt{1+49}\\\\d = \sqrt{50}\\\\d \approx 7.07107\\\\[/tex]

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To summarize, we have these approximate segment lengths.

  • PQ = 6.403
  • PR = 7.071

Segment PQ is shorter, which means Q is the closer point.