Determine Whether The Point Is Collinear. (1) A(1, –3), B(2, –5), C

English0Maharashtra State BoardSSC (English Medium) 10th StandardQuestion PapersQuestion Papers369Textbook Solutions22378MCQ Online Mock Tests39Important Solutions7182Concept Notes & Videos454Time Tables31SyllabusDetermine whether the points are collinear. A(1, −3), B(2, −5), C(−4, 7) - Geometry Mathematics 2

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Questions

Determine whether the points are collinear.

A(1, −3), B(2, −5), C(−4, 7)

Verify, whwther points, A(1, −3), B(2, −5) and C(−4, 7) are collinear or not.

Sum

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SolutionShow Solution

Given: A(1, −3), B(2, −5), C(−4, 7)

Let,

A(1, −3) = A(x1, y1)

B(2, −5) = B(x2, y2)

C(−4, 7) = C(x3, y3)

By the distance formula,

d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

= `sqrt((2 - 1)^2 + [-5 - (-3)]^2)`

= `sqrt((1)^2 + (-5 + 3)^2)`

= `sqrt((1)^2 + (-2)^2)`

= `sqrt(1+ 4)`

= `sqrt(5)` ...(1)

d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2)`

= `sqrt((- 4 - 2)^2 + [7 - (-5)]^2)`

= `sqrt((-6)^2 + [7 + 5]^2)`

= `sqrt((-6)^2 + (12)^2)`

= `sqrt(36 + 144)`

= `sqrt(180)`

= `sqrt(36 xx 5)`

= `6sqrt(5)` ...(2)

d(A, C) = `sqrt((x_3 - x_1)^2 + (y_3 - y_1)^2)`

= `sqrt((-4 - 1)^2 + [7 - (-3)]^2)`

= `sqrt((-4 - 1)^2 + (7 + 3)^2)`

= `sqrt((-5)^2 + (10)^2)`

= `sqrt(25 + 100)`

= `sqrt(125)`

= `sqrt(25 × 5)`

= `5sqrt(5)` ...(3)

Adding (1) and (3)

∴ d(A, B) + d(A, C) = d(B, C)

∴ `sqrt5 + 5sqrt5 = 6sqrt5` ...(4)

∴ d(A, B) + d(A, C) = d(B, C) ...[From (2) and (4)]

∴ Points A(1, −3), B(2, −5) and C(−4, 7) are collinear. ...(Hence proved)

shaalaa.comDistance Formula Report Error Is there an error in this question or solution?Q 2.1Q 1.6Q 2.2Chapter 5: Co-ordinate Geometry - Practice Set 5.1 [Page 107]

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Balbharati Geometry (Mathematics 2) [English] 10 Standard SSC Maharashtra State BoardChapter 5 Co-ordinate GeometryPractice Set 5.1 | Q 2.1 | Page 1072019-2020 (March) Shaalaa.com Model Set 2 (with solutions)Q 2.B.ii | 2 marks2023-2024 (March) Official (with solutions)Q 3.B.1 | 3 marks2024-2025 (March) Official (with solutions)Q 2.B.4 | 2 marks

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