Geometry - Prove that these 3 points don't lie on the same line

Question

Rectangular coordinate system in the plane The Points are given - $A(5,0) B(1,3) C(-2,0)$. Prove that these $3$ points don't lie on the same line and find the surface of the triangle $ABC$.

Answer 1

You don't usually solve problems like this by "getting a formula", you solve them by thinking.

Draw a picture of the coordinate plane, put a dot at each of the three points. See whether they are on a line.

Then you should be able to say in just a few words why $B$ is not on the line containing $A$ and $C$.

Finally, you can use a formula to find the area of the triangle.

Answer 2

In the comments, you asked for a formula. This is a bad idea. The purpose of mathematics is to understand, not to plug numbers into formulas, and we are not computers.

So let me give some steps you can take to solve the problem.

1. Draw a pictures! Take some graph paper, or make a grid on some ordinary paper. Find the points $(5,0)$, $(1,3)$, and $(-2, 0)$. Color them in. Do they appear to lie on a straight line?
2. Do they look like they form a triangle? What do you know about triangles and their areas? Do you know the formula for area of a triangle - do you understand why it is true?
3. What would the area of their triangle be if they all were on a straight line? Perhaps finding their area can help you solve your first problem as well!

Answer 3

Try this for the triangle (0,0)(a,b)(c,d) Area=1/2 [ ad-bc]

Change your coordinates (5,0)----->(0,0) (-2,0)--->(-7,0) (1,3)----->(-4,3) Area=1/2 [ -21+0]=21/2 answer