Find the length of each side of the triangle determined by the three points P1,P2, and P3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.)
P1=(−1,4);P2=(6,2);P3=(4,−5)

The triangle is both an Isosceles triangle and a right triangle.

Step-by-step explanation:

Given the vertices of a triangle.

   and

We find the distance between all the points to determine the length of each side of the triangle.

Distance between any two points, say, and is:

Length between and , (Side 1) :

    and

Distance =

Length of Side 1 = units.

Distance between and , (Side 2):

Distance =

Length of Side 2 = units.

Distance between and , Side 3 :

Distance =

Length of Side 3  = units.

Note that the length of Side 1 = Length of Side 3.

That means the triangle is Isosceles.

Also, For a triangle to be right angle triangle, using Pythagoras theorem we have:

(Side 1)² + (Side 3)² = (Side 2)²

i.e, 53 + 53  = 106

Hence, the triangle is a right - angled triangle as well.

6e14 is the answer.