A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem. A triangle, drawn with side lengths equivalent to a Pythagorean triple, will be a right triangle.
Here are two examples of Pythagorean triples.
Can you find another Pythagorean triple?
How many do you think there are?
Is there any easy way to find an infinite number of triples?
Given a Pythagorean triple, we can find another set by multiplying by a whole number to make a similar triangle with proportional sides.
The Pythagorean theorem and its converse describe how the side lengths of right triangles are related.
If a triangle is not a right triangle, then the Pythagorean inequality theorem can be used to determine the type of triangle based on angles. Where c is the longest side, we have that:
If c^2<a^2+b^2, then the triangle is acute
If c^2>a^2+b^2, then the triangle is obtuse
Find the value of c for each triangle.
Select all of the Pythagorean triples.
For the given triangle:
Is the triangle a right triangle? Explain.
Classify the triangle in terms of its sides and angles.
Archeologists have uncovered an ancient pillar which, after extensive digging, remains embedded in the ground. The lead researcher wants to record all of the dimensions of the pillar, including its height above the ground.
However, the team can only take certain measurements accurately without risking damage to the artifact. These measurements are shown in the diagram.
Find the value of the variables.
Given two sides in a right triangle, we can find the third side length using the Pythagorean theorem.
Pythagorean theorem: If a triangle is a right triangle, then the square of the length of the hypotenuse is equal to the sum of the squares of its leg lengths
The Pythagorean inequality theorem can be used to determine if a non-right triangle is acute or obtuse.
Converse of the Pythagorean theorem: If the lengths a, b and c of the three sides of a triangle satisfy the relationship a^2+b^2=c^2, then the triangle is a right triangle
If a,\,b,\, and c satistfy a^2+b^2=c^2, and are positive integers, then we say that they form a Pythagorean triple. Any whole number scalar multiple of a Pythagorean triple is also a Pythagorean triple.