Prove that the triangle with vertices with (3,5), (-2,6), and (1,3) is a right triangle.

The triangle with vertices with (3, 5), (-2, 6), and (1, 3) is a right triangle.

Solution:

Given that the vertices of triangle are (3, 5), (-2, 6) and (1, 3)

Let us consider A(3, 5) B(-2, 6) C(1, 3)

If the sum of square of distance between two vertices is equal to the square of distance between third vertices, then the triangle is a right angled triangle.

By above definition, we get  

----- eqn 1

Where AB is the distance between vertices A and B

BC is the distance between vertices B and C

CA is the distance between vertices C and D

Distance between any two vertices of a triangle is given as  

------- eqn 2

Step 1:

Let us find the distance between the vertices A(3,5) and B(-2,6)  

By using equation 2, we get  

Distance between vertices A and B =

=

=

=

=

Step 2:

Let us find the distance between the vertices B(-2,6) and C(1,3)

By using equation 2, we get  

Distance between vertices B and C =

=

=

=

=

Step 3:

Let us find the distance between the vertices C(1,3) and D(3,5)

By using equation 2, we get  

Distance between the vertices C and A

=

=

=

=

Step 4:

By using equation 1,

18 + 8 = 26

26 = 26

Hence the condition is satisfied. So the given triangle with vertices with (3,5), (-2,6), and (1,3) is a right triangle.  

answer are

step-by-step explanation: