MHB Converse of Pythagorean Theorem

[mathdad]

Determine if the triangle with the given vertices is a right triangle.

(7, -1), (-3, 5), (-12, -10)

I must find the lengths of the sides using the distance formula for points on the xy-plane.

The question then tells me to use the converse of the Pythagorean theorem.

How do I use this theorem? What does the theorem say?

 

[Greg]

The Pythagorean theorem states that, given a right-angled triangle, the sum of the squares of the lengths of the two legs is equal to the square of the hypotenuse. That is, if the legs are a and b and the hypotenuse is c then

$$a^2+b^2=c^2$$

The converse is also true; that is, if the lengths of two sides of a triangle, when squared and added together, are equal to the square of the length of the third side then the triangle is right angled.

 

[MarkFL]

Alternately, since no point shares a corresponding coordinate with any other, you can check to see if any of the 3 possible pairs of line segments have slopes whose product is -1. :D

 

[I like Serena]

Just a fun fact, the reverse of the Pythagorean is a way that builders ensure that walls are properly perpendicular to each other.
Put up strings of 3, 4, and 5 meters in a triangle, and we can build perpendicular walls along those strings.
Or 30, 40, and 50 meters for that matter.
We can't do that with a measuring triangle, since that is too small.

 

[mathdad]

After finding the lengths of all 3 sides, I can let a = length 1, b = length 2 and c = length 3.

(length 1)^2 + (length 2)^2 = (length 3)^2

Correct?

 

[HallsofIvy]

After finding the lengths of all 3 sides, I can let a = length 1, b = length 2 and c = length 3.

(length 1)^2 + (length 2)^2 = (length 3)^2

Correct?

It helps to know that the hypotenuse, if this is a right triangle, is always the largest of the three sides and that which of the other two sides you call "a" and "b" is irrelevant so you know which to call "a", "b", and "c".

 

[mathdad]

Should it be |a|^2 + |b|^2 = |c|^2?

I am going to work it out and post my complete work when time allows.

 

[MarkFL]

Should it be |a|^2 + |b|^2 = |c|^2?

I am going to work it out and post my complete work when time allows.

a, b and c will be distances obtained from using the distance formula, and so will be non-negative to begin with. :D

 

[mathdad]

Thank you everyone.