Find the distance between the following pairs of points:
(i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)

2 thoughts on “Find the distance between the following pairs of points:<br /> (i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)”

1. Answer:

   [tex]\bf\red{Question}[/tex]

   Find the distance between the following pairs of points:

   (i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)

   [tex]\bf\red{To\:find}[/tex]

   We have to find the distance between following pair of points

   [tex]\bf\red{Formula}[/tex]

   [tex] \sqrt{(x_ 2 – x_1) {}^{2} + (y_2 – y_1) } {}^{2} [/tex]

   [tex]\bf\red{Solution}[/tex]

   (i) (2, 3), (4, 1)

   [tex] \sqrt{( 2 – 4) {}^{2} + (3 – 1 } {}^{2} [/tex]

   [tex] \sqrt{(2 – 2) {}^{2} + (2) {}^{2} } = \sqrt{4 + 4 }= \sqrt{8} = 2 \sqrt{2} [/tex]

   (ii) (–5, 7), (–1, 3)

   [tex] \sqrt{( – 5 – ( – 1)) {}^{2} + (7 – 3) {}^{2} } [/tex]

   [tex] \sqrt{( – 4) {}^{2} + (4) {}^{2} } = \sqrt{16 + 16} = \sqrt{32} = 4 \sqrt{2} [/tex]

   (iii) (a, b), (–a, –b)

   [tex] \sqrt{(a – ( – a)) {}^{2} + (b – ( – b)) {}^{2} } [/tex]

   [tex] \sqrt{(2a) {}^{2} + (2b) {}^{2} } = \sqrt{4a {}^{2} + 4 {b}^{2} } = 2 \sqrt{a {}^{2} + {b}^{2} } [/tex]

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2. Solutions:-

   To Find:-

   • Distance between the following points

   (i) (2, 3) and (4, 1)

   We know the distance formula is as follows:-

   • [tex]\dag{\boxed{\underline{\pink{\rm{\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}}}}}}[/tex]

   Where:-

   • x₁ = abscissa of the first coordinate
   • x₂ = abscissa of the second coordinate
   • y₁ = ordinate of the first coordinate
   • y₂ = ordinate of the second coordinate

   Here,

   Let the points on both the Coordinates be P and Q, i.e. P(2, 3) and Q(4, 1), Such that:-

   [tex]\sf{x_1 = 2\:and\:x_2 = 4}[/tex]

   [tex]\sf{y_1 = 3\:and\:y_2 = 1}[/tex]

   Putting all the values in the formula:–

   [tex]\sf{PQ = \sqrt{(4-2)^2 + (1-3)^2}}[/tex]

   [tex] = \sf{PQ = \sqrt{(2)^2 + (-2)^2}}[/tex]

   [tex] = \sf{PQ = \sqrt{4 + 4}}[/tex]

   [tex]= \sf{PQ = \sqrt{8}}[/tex]

   [tex] = \sf{PQ = 2\sqrt{2}}[/tex]

   ∴ The distance between the points P(2, 3) and Q(4, 1) is 2√2 units.

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   (ii) (-5, 7) and (-1, 3)

   ⟶ Let A and B be the points on coordinates (-5, 7) and (-1, 3) respectively, such that:-

   [tex]\sf{x_1 = -5\:and\:x_2 = -1}[/tex]

   [tex]\sf{y_1 = 7\:and\:y_2 = 3}[/tex]

   Putting all the values in the formula:–

   [tex]\sf{AB = \sqrt{[(-1) – (-5)]^2 + (3 – 7)^2}}[/tex]

   [tex] = \sf{AB = \sqrt{(-1 + 5)^2 + (-4)^2}}[/tex]

   [tex] = \sf{AB = \sqrt{(4)^2 + 16}}[/tex]

   [tex] = \sf{AB = \sqrt{16 + 16}}[/tex]

   [tex] = \sf{AB = \sqrt{32}}[/tex]

   [tex] = \sf{AB = 4\sqrt{2}}[/tex]

   ∴ The distance between the points A(-5, 7) and B(-1, 3) is 4√2 units.

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   (iii) (a, b) and (-a, -b)

   ⟶ Let C and D be the points on coordinates (a, b) and (-a, -b) respectively, such that:+

   [tex]\sf{x_1 = a\:and\:x_2 = -a}[/tex]

   [tex]\sf{y_1 = b\:and\:y_2 = -b}[/tex]

   Putting all the values in the formula:–

   [tex]\sf{CD = \sqrt{(-a – a)^2 + (-b – b)^2}}[/tex]

   [tex] = \sf{CD = \sqrt{(-2a)^2 + (-2b)^2}}[/tex]

   [tex] = \sf{CD = \sqrt{4a^2 + 4b^2}}[/tex]

   [tex] = \sf{CD = \sqrt{4(a^2 + b^2)}}[/tex]

   [tex] = \sf{CD = 2\sqrt{a^2 + b^2}}[/tex]

   ∴ The distance between the points C(a, b) and D(-a, -b) is 2√a² + b² units.

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