Question

Using Slopes Verify the given geometric property.

Use slopes to determine whether the given points are collinear (lie on a line).

(a)$\left(1,1\right),\left(3,9\right),\left(6,21\right)$

(b) $\left(-1,13\right),\left(1,7\right),\left(4,15\right)$are vertices of a rectangle.

Short answer

1. The points are collinear.
2. The given points are not collinear.

Step-by-step solution

Step 1. Concept of collinear point.

Collinear points are the set of three or more points that exist on the same straight line that is slope of each pair of points are same.

Step 2. Answer the question (a).

Let the points are $A\left(1,1\right),B\left(3,9\right),C\left(6,21\right)$.

The slope of$AB=\frac{3-1}{9-1}=\frac{2}{8}=\frac{1}{4}=m_1$ (say).

The slope of$BC=\frac{6-3}{21-9}=\frac{3}{12}=\frac{1}{4}=m_2$ (say).

The slope of$AC=\frac{6-1}{21-1}=\frac{5}{20}=\frac{1}{4}=m_3$ (say).

Step 2. Answer of question (b).

Let the points are $A\left(-1,3\right),B\left(1,7\right),C\left(4,15\right)$.

The slope of $AB=\frac{1+1}{7-3}=\frac{2}{4}=\frac{1}{2}=m_1$(say).

The slope of $BC=\frac{4-1}{5-7}=\frac{3}{-2}=m_2$(say).

The slope of $AC=\frac{4+1}{5-3}=\frac{5}{2}=m_3$(say).

Since,$m_1\ne m_2\ne m_3$ that is points $\left(-1,13\right),\left(1,7\right),\left(4,15\right)$are not collinear.