SOLUTION: Q.1: Find the lenghts of the sides of the triangle with vertices A(-4,0), B(3,4) and C(4,1).show that the triangle ABC is isosceles?

There are two ways. It will be instructive to learn
both methods.

Find the length of AB by drawing this blue
right triangle with AB as its hypotenuse: 



The bottom leg of the blue triangle is obviously 7 units long.
(The bottom leg looks purple since it's blue over a red x-axis).
You can just count them. And the right leg of the blue triangle 
is obviously 4 units tall.  You can tell by looking at the units 
of the y axis.

Now AB is the hypotenuse of the blue right triangle. So we use
the Pythagorean theorem:






Now we erase the blue right triangle and draw a
green one:

We find the length of AC by drawing this green
right triangle with AC as its hypotenuse:

 

The bottom leg of the green triangle is obviously 8 units long.
And the right leg of the green triangle is obviously 1 unit tall.

Now AC is the hypotenuse of the green right triangle. So we use
the Pythagorean theorem:






So the lengths of AB and AC are both equal to.
Therefore triangle ABC is isosceles.

HERE'S THE SECOND WAY;

Use the distance formula to find AB:








Use the distance formula to find AC:








So the lengths of AB and AC are both equal to.
Therefore triangle ABC is isosceles.

Edwin