Question 3 of 18intermediate🔧 ApplyShort Answer2 marks

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

Correct Answer

Let the given points be A(1, 5), B(2, 3), and C(-2, -11). To determine if the points are collinear, we calculate the distances between them using the distance formula.

First, we calculate the distance AB: AB = √[(2 – 1)² + (3 – 5)²] = √[(1)² + (–2)²] = 1+4\sqrt{1 + 4} = 5\sqrt{5}.

Next, we calculate the distance BC: BC = √[(–2 – 2)² + (–11 – 3)²] = √[(–4)² + (–14)²] = 16+196\sqrt{16 + 196} = 212\sqrt{212}.

Finally, we calculate the distance AC: AC = √[(–2 – 1)² + (–11 – 5)²] = √[(–3)² + (–16)²] = 9+256\sqrt{9 + 256} = 265\sqrt{265}.

For the points to be collinear, the sum of two distances must equal the third (AB + BC = AC). Here, AB + BC = 5\sqrt{5} + 212\sqrt{212}. Since 5\sqrt{5} + 212\sqrt{212} ≠\neq 265\sqrt{265} (as 5 + 212 = 217 ≠\neq 265), the points are not collinear.

Exercise: EXERCISE 7.1 | Q: 3 | (Page 7)
For More Understanding

Explanation

The textbook context demonstrates that points are collinear if the sum of the distances between two pairs of points equals the distance between the third pair (AB + BC = AC). Applying the distance formula to the points (1, 5), (2, 3), and (-2, -11), we find the distances are 5\sqrt{5}, 212\sqrt{212}, and 265\sqrt{265}. Since the sum of the squares of the smaller distances (5 + 212 = 217) does not equal the square of the largest distance (265), the condition AB + BC = AC fails.

Solution Steps

  1. Step 1: Calculate distance AB = √[(2-1)² + (3-5)²] = 1+4\sqrt{1 + 4} = 5\sqrt{5}.

  2. Step 2: Calculate distance BC = √[(-2-2)² + (-11-3)²] = 16+196\sqrt{16 + 196} = 212\sqrt{212}.

  3. Step 3: Calculate distance AC = √[(-2-1)² + (-11-5)²] = 9+256\sqrt{9 + 256} = 265\sqrt{265}.

  4. Step 4: Compare AB + BC with AC. Since 5\sqrt{5} + 212\sqrt{212} ≠\neq 265\sqrt{265}, the points are not collinear.

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