Chapter 1: Orienting Yourself: The Use of Coordinates

Exercise

Question

Question: 7

9 / 25

Use your method (from Problem 6) to check if the points R (-5, -1), B (-2, -5) and C (4, -12) are on the same straight line.

Now plot both sets of points and check your answers.

Answer

1
Find the distances between each pair of points using the Baudhāyana-Pythagoras theorem.
RB = \sqrt{(-2 - (-5))^2 + (-5 - (-1))^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
BC = \sqrt{(4 - (-2))^2 + (-12 - (-5))^2} = \sqrt{6^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85} \simeq 9.22
CR = \sqrt{(-5 - 4)^2 + (-1 - (-12))^2} = \sqrt{(-9)^2 + 11^2} = \sqrt{81 + 121} = \sqrt{202} \simeq 14.21
2
Compare the sum of the two smaller distances with the largest distance. If the sum is equal to the largest distance, the three points are collinear.
Answer: No. The points R, B, and C do not lie on the same straight line.
\textbf{Reason: The two smaller distances are RB = 5 and BC = $\sqrt{85}$ $\simeq$ 9.22, while the largest distance is CR = $\sqrt{202}$ $\simeq$ 14.21. Their exact sum is 5 + $\sqrt{85}$, which is not equal to $\sqrt{202}$. Therefore, R, B, and C are not collinear.}
3
Plot both sets of points and join each set in the given order. Check whether the three points in each set lie on the same straight line.
Answer: The graph confirms our results.
Reason: M, A, and G lie on the same straight line, while R, B, and C do not. The graph confirms the results obtained using the distance method.
Coordinate plane showing points M, A, and G on one straight line, while points R, B, and C do not lie on the same straight line.
Graph verifying that M, A, and G are collinear, while R, B, and C are not.
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