Question

Prove that (2, -2), (-2, 1) and (5, 2) are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.

Answer 1

Answer:

.................... hi

Answer 2

Here to calculate the area of the triangle we have to determine if the triangle is right angled.

Explanation:

Let us focus on two perpendicular lines in triangle ABC.

A = (2,-2)

B = (-2, 1)

C = (5,2)

Here AC and AB are two perpendicular lines. So if the lines in a slope are perpendicular, then the slope product should be -1.

Slope of AB = 1 – (-2) / (-2) –(2)

 1 + 2 / -4 = 3 / -4.

Slope of AC = (-2) -2 / 2 – 5

 -4 / -3 = 4/3.

So the product of two slopes = -3/4 x 4/3 = -1.

So the two lines are perpendicular.  

This means that the triangle is right angled at A.

Now to find the area of the right angled triangle we need to find the length

Slope of line AB = -3/4.

Slope of line AC = 4/3

Thus area of the triangle = ½ b h

  = ½ (-3/4) (4/3)

  = ½ (-1)

  -1/2.

Now to  calculate the length of the hypotenuse we need the slope of BC

= 1 – 2 / -2 – (-5)

= -1 / 2+5

= -1 / 7.

(figure attached).