Q. 13 - Geometry

Question

Given points A, B, and C. Find AB, BC, and AC. Are A, B, and C collinear?

If so, which point lies between the other two?

13. A(0, 3), B(-2, 1), C(3, 6)

Step-by-Step Solution

Answer

The values of$AB=2\sqrt{2},BC=5\sqrt{2}\text{ , }AC=3\sqrt{2}$ .

 A, B, C are collinear. 

Point A lies between the other two points.

Step-1 – Given

The given points are $A\left(0,3\right)\text{ , }B\left(-2,1\right)\text{ , }C\left(3,6\right)$

Step-2 – To determine

We have to find the lengths AB, BC and AC. Then have to check if A, B, C are collinear. And determine that which point lies between the other two.

Step-3 – Calculation

Using the distance formula, the distance between A and B is:

$\begin{array}{l}AB=\sqrt{{\left(-2-0\right)}^2+{\left(1-3\right)}^2}\\ AB=\sqrt{{\left(-2\right)}^2+{\left(-2\right)}^2}\\ AB=\sqrt{4+4}\\ AB=\sqrt{8}\\ AB=2\sqrt{2}\end{array}$

Using the distance formula, the distance between B and C is:

$\begin{array}{l}BC=\sqrt{{\left(3-\left(-2\right)\right)}^2+{\left(6-1\right)}^2}\\ BC=\sqrt{{\left(3+2\right)}^2+{\left(6-1\right)}^2}\\ BC=\sqrt{{\left(5\right)}^2+{\left(5\right)}^2}\\ BC=\sqrt{25+25}\\ BC=\sqrt{50}\\ BC=5\sqrt{2}\end{array}$

Using the distance formula, the distance between A and C is:

$\begin{array}{l}AC=\sqrt{{\left(3-0\right)}^2+{\left(6-3\right)}^2}\\ AC=\sqrt{{\left(3\right)}^2+{\left(3\right)}^2}\\ AC=\sqrt{9+9}\\ AC=\sqrt{18}\\ AC=3\sqrt{2}\end{array}$

Hence,

$\begin{array}{l}AB+AC=BC\\ BC=2\sqrt{2}+3\sqrt{2}\\ BC=5\sqrt{2}\end{array}$

So, the values of $AB=2\sqrt{2},BC=5\sqrt{2}\text{ , }AC=3\sqrt{2}.$

A, B, C are collinear. 

Point A lies between the other two points.