Area - Triangle, Given Three Vertices, 3

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Show that the points (5, 4), (-2, 1), and (2, -3) are the vertices of an isosceles triangle, and find its area.

Solution: 

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The first thing that we need to do is to get the length of the sides of a triangle using the distance of two points as follows

For the length of AB:

For the length of AC:

For the length of BC:

Since AB ≅ AC, then the given triangle is an isosceles triangle. 

The area of a triangle is given by the formula

Substitute the values of the coordinates of the vertices of a triangle to the above equation and solve for the value of matrix or determinant, we have

Therefore, the area of a triangle is

 

Area - Triangle, Given Three Vertices, 2

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Show that the points (-2, 0), (2, 3), and (5, -1) are the vertices of a right triangle, and find its area.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

The first thing that we need to do is to get the slope of each sides of a triangle using the two point formula as follows

For the slope of AB:

For the slope of BC:

For the slope of AC:

Since the slope of AB is negative reciprocal of the slope of BC, then the given triangle is a right triangle. A right triangle is a triangle whose two sides are perpendicular . 

The area of a triangle is given by the formula

Substitute the values of the coordinates of the vertices of a triangle to the above equation and solve for the value of matrix or determinant, we have

The area of any closed plane figure is always in absolute value. In this case, if the calculated area is negative, then change the negative sign into positive sign. Therefore, the area of a triangle is