Converse of Pythagorean Theorem

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Discussion Overview

The discussion centers on determining whether a triangle defined by specific vertices is a right triangle, utilizing the converse of the Pythagorean theorem. Participants explore the theorem's implications, methods of verification, and related concepts, including practical applications.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on how to apply the converse of the Pythagorean theorem to determine if the triangle is right-angled.
  • Another participant explains the Pythagorean theorem and its converse, stating that if the sum of the squares of two sides equals the square of the third side, the triangle is right-angled.
  • Some participants suggest checking the slopes of line segments formed by the triangle's vertices to determine if they are perpendicular.
  • A participant mentions a practical application of the Pythagorean theorem in construction, illustrating how it can be used to ensure walls are perpendicular.
  • Several participants discuss the assignment of side lengths to variables a, b, and c, with some confusion about whether to use absolute values in the equation.
  • One participant expresses intent to work through the problem and share their findings later.

Areas of Agreement / Disagreement

Participants generally agree on the application of the Pythagorean theorem and its converse, but there is some uncertainty regarding the assignment of side lengths and whether absolute values should be used. The discussion remains unresolved regarding the correct approach to applying these concepts to the specific triangle.

Contextual Notes

Participants have not yet provided complete calculations or definitive conclusions about the triangle's status as a right triangle. The discussion includes various assumptions about side lengths and their relationships.

mathdad
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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?
 
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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.
 
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
 
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.
 
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?
 
RTCNTC said:
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".
 
Should it be |a|^2 + |b|^2 = |c|^2?

I am going to work it out and post my complete work when time allows.
 
RTCNTC said:
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
 
Thank you everyone.
 

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