Question

2. Determine whether the points are collinear.
(1) A(1.-3), B(2.-5), C(-4,7)
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Answer 1

Question

Determine whether the points are colinear.

Method

Let's assume point A lies inside $\overline{BC}$.

Then, it should satisfy $\overline{AB} + \overline{CA} = \overline{BC}$.

Each length of sides:

• $\overline{AB}^2 = 1^2 + 2^2 \implies \overline{AB} = \sqrt{5}$
• $\overline{BC}^2 = 6^2 + 12^2 \implies \overline{BC} = 6\sqrt{5}$
• $\overline{CA}^2 = 5^2 + 10^2 \implies \overline{CA} = 5\sqrt{5}$

The equation is satisfied. Hence, point A lies inside $\overline{BC}$. And therefore, three points are colinear.

More information

Triangle Inequality

Three sides of a triangle must satisfy that sum of the remaining sides are longer than the longest side.

Let's say $\overline{CA}$ is the longest side, and one of the vertices is $B$.

$\implies \overline{AB} + \overline{BC} \geq \overline{CA}$

$\mathrm{\implies \overline{AB} + \overline{BC} \geq \overline{CA} \: [\overline{AB} + \overline{BC} = \overline{CA} \: where \: B \in \overline{CA}]}$

Answer 2

Step-by-step explanation:

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