This question is pertaining to College Algebra

Question

Determine whether the three points are the vertices of a  right triangle:

 

(-7, 4), (6, -2), (0, -15)

 

Determine whether the three points are collinear

 

(-4, 3), (2,5), (-1, 4)

Answer 1

Hi Keissa.

 

To find out if the three points are the vertices of a right triangle, let's first determine the lengths of the three sides they create using the distance formula.

 

I'll define the points as follows:

 

A(-7,4)

B(6,-2)

C(0,15)

 

The distance formula is:

 

D = √((x₂ - x₁)² + (y₂ - y₁)²)

 

Let's get started.

 

AB = √((6- -7)² + (-2-4)²)

 

= √(169 + 36)

 

= √205

 

BC = √((0-6)² + (15- -2)²)

 

= √(36 + 289)

 

= √(325)

 

AC = √((0 - -7)² + (15 - 4)²)

 

= √(49 + 121)

 

= √(170)

 

Take a look at your largest length. It happens to be BC.

 

If you have a right triangle, then BC would be your hypotenuse, and:

 

AB² + AC² = BC²

 

√(205)² + √(170)² = √(325)²

 

375 = 325

 

Nope. Not a right triangle.

 

**

 

In order to figure out if 3 points are collinear, find the equation of the line for two of them, then plug in the third point and see if the equation fits.

 

Let's find the equation of the line through the first 2 points:

 

(-4, 3)

(2, 5)

 

m = (5 - 3)/(2 - -4)

 

m = 2/6 = 1/3

 

y - y₁ = m(x - x₁)

 

y - 5 = 1/3(x - 2)

 

y - 5 = 1/3x - 2/3

 

y = 1/3x +13/3

 

Cool. Now let's plug in the third point and see if it works.

 

4 = 1/3(-1) + 13/3

 

4 = -1/3 + 13/3

 

4 = 12/3

 

4 = 4

 

Yep! They're collinear.

 

Hope this helps.