Distance in the plane

If (a,b) and (c,d) are the coordinates of two points in the plane, it follows from the Pythagorean Theorem that the distance between these point is  

$$\sqrt{(a-c)^2 + (b-d)^3}.$$ (1)

Example: If A has coordinates (-1,2) and B has coordinates (2,7) then the distance from A to B is

$$\sqrt{(-1-2)^2 + (2-7)^2} = \sqrt{9 + 25} = \sqrt{34}$$

Exercises

1.

Find the distance between each pair of points:

(a)

(1,2) and (3,4);

(b)

(-11,4) and (10,14);

(c)

(3,-5) and $(-8,\sqrt{2})$;

2.

Do the points (2,4), (4,9), and (-5,2) form the vertices of a right triangle? Explain.

3.

Draw a graph to illustrate the inequality $\sqrt{(x-2)^2 + (y-3)^2} \leq 4$.

4.

Draw a graph to illustrate the inequality $\sqrt{(x+2)^2 + (y-3)^2} \leq 4$.