Determine if the points and are collinear.

Coordinate Geometry | Exercise 7.1

Question 3

Determine if the points $(1, 5), (2, 3)$ and $(-2, -11)$ are collinear.

Solution

Understand the Question

Three points $A$ , $B$ , and $C$ are collinear if they lie on the same straight line, which requires the sum of the lengths of any two segments to equal the length of the third segment (e.g., $AB + BC = CA$ ).
The distance formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
We find the distances $AB$ , $BC$ , and $CA$ , and verify whether the sum of any two distances equals the third.

Step 1 · Calculate Distance AB

Let the given points be $A(1, 5)$ , $B(2, 3)$ , and $C(-2, -11)$ .Using the distance formula for $A(1, 5)$ and $B(2, 3)$ :

\begin{aligned} AB &= \sqrt{(2 - 1)^2 + (3 - 5)^2} \\[0.6em] &= \sqrt{(1)^2 + (-2)^2} \\[0.6em] &= \sqrt{1 + 4} \\[0.6em] &= \sqrt{5} \end{aligned}

Step 2 · Calculate Distance BC

Using the distance formula for $B(2, 3)$ and $C(-2, -11)$ :

\begin{aligned} BC &= \sqrt{(-2 - 2)^2 + (-11 - 3)^2} \\[0.6em] &= \sqrt{(-4)^2 + (-14)^2} \\[0.6em] &= \sqrt{16 + 196} \\[0.6em] &= \sqrt{212} \end{aligned}

Step 3 · Calculate Distance CA

Using the distance formula for $C(-2, -11)$ and $A(1, 5)$ :

\begin{aligned} CA &= \sqrt{(1 - (-2))^2 + (5 - (-11))^2} \\[0.6em] &= \sqrt{(1 + 2)^2 + (5 + 11)^2} \\[0.6em] &= \sqrt{(3)^2 + (16)^2} \\[0.6em] &= \sqrt{9 + 256} \\[0.6em] &= \sqrt{265} \end{aligned}

Step 4 · Check for Collinearity

Comparing the calculated distances:

\begin{aligned} AB + BC &= \sqrt{5} + \sqrt{212} \approx 2.236 + 14.560 = 16.796 \\[0.6em] CA &= \sqrt{265} \approx 16.279 \end{aligned}

Since $16.796 \neq 16.279$ , $AB + BC \neq CA$ .

Checking the other combinations: $AB + CA = \sqrt{5} + \sqrt{265} \approx 2.236 + 16.279 = 18.515 \neq BC$ $BC + CA = \sqrt{212} + \sqrt{265} \approx 14.560 + 16.279 = 30.839 \neq AB$

Since no sum of two segment lengths equals the third, the points are not collinear.

Answer

The points $(1, 5)$ , $(2, 3)$ , and $(-2, -11)$ are not collinear.

Common Mistakes

Adding Under the Square Root: Incorrectly adding square roots as $\sqrt{5} + \sqrt{212} = \sqrt{217}$ . Surds with different radicands cannot be directly added together under a single radical.
Sign Errors with Negative Coordinates: Mishandling double negatives during subtraction, such as writing $1 - (-2)$ as $1 - 2 = -1$ instead of $1 + 2 = 3$ .
Incomplete Verification: Only checking $AB + BC = CA$ without identifying which segment is the longest.

Question 3

More questions in Exercise 7.1