Question

Critical Thinking A primitive Pythagorean triple is a set of whole numbers that satisfies the equation \(a^{2}+b^{2}=c^{2}\) and has no common factors except 1. A family of Pythagorean triples is a primitive triple and its whole number multiples. How are the triangles represented by a family of Pythagorean triples related? Explain.

Short answer

Triangles in a family of Pythagorean triples are similar; they have identical angle measures and proportional sides.

Step-by-step solution

Understanding Primitive Pythagorean Triples

Primitive Pythagorean triples are sets of three whole numbers \(a\), \(b\), and \(c\) that satisfy the equation \(a^2 + b^2 = c^2\), where \(a\), \(b\), and \(c\) have no common divisor other than 1. Examples include (3, 4, 5) and (5, 12, 13).

Understanding Family of Pythagorean Triples

A family of Pythagorean triples includes a primitive Pythagorean triple and all of its multiples by any positive integer. For instance, multiplying the primitive triple (3, 4, 5) by 2 gives another triple (6, 8, 10), which is part of this family.

Relationship Among the Triangles

The triangles represented by a family of Pythagorean triples are similar to each other. Since the sides of these triangles are multiples of one another, they maintain the same angle measures and thus form similar triangles based on the criteria for triangle similarity (AA, SAS, or SSS).

Analyzing the Side Ratios

All triangles in the family maintain the same side ratios, meaning that the ratios \(\frac{a}{b}, \frac{b}{c},\) and \(\frac{c}{a}\) remain constant across the family at integer multiples. Hence, multiplying all sides of the primitive triple by an integer affects the size but not the shape of the triangle.

Key concepts

triangle similarity

Triangle similarity is a fundamental concept in geometry that occurs when two triangles have the same shape but not necessarily the same size. This similarity is established when triangles have equal corresponding angles and proportional corresponding sides. There are three main criteria to determine if two triangles are similar: Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS).

In the context of Pythagorean triples, when you have a set of Pythagorean triples such as (3, 4, 5), and its multiples like (6, 8, 10), all the triangles formed by these triples are similar. This is because multiplying the sides by the same integer maintains the angles, ensuring the triangles have the same shape.

Understanding triangle similarity helps to grasp why families of Pythagorean triples result in similar triangles. The side lengths remain in proportion, which guarantees that the triangles will always be similar, maintaining constant angle measures no matter how large or small the multiple.

primitive Pythagorean triples

Primitive Pythagorean triples are special sets of three whole numbers that fulfill the equation \(a^2 + b^2 = c^2\), where the greatest common divisor of these numbers is 1. This means these numbers have no common factor other than 1. Typical examples of such triples include (3, 4, 5) and (5, 12, 13).

What makes primitive Pythagorean triples fascinating is their role in forming the basic building blocks of all possible Pythagorean triples. By multiplying a primitive triple by any positive integer, you generate a family of triples, all related to the original primitive set. This multiplication yields new triples such as (6, 8, 10), all of which maintain the properties of Pythagorean triples.

The origin of these primitive triples can be traced back to ancient mathematics and they are still extensively used in modern mathematical applications involving geometry and number theory.

Pythagorean theorem

The Pythagorean theorem is an essential principle in mathematics, specifically in geometry. It defines a fundamental relationship in right-angled triangles. According to the theorem, in any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.

This theorem not only helps in solving problems related to right triangles but is also the basis for identifying Pythagorean triples. A set of numbers forms a Pythagorean triple if they can be the sides of a right triangle and satisfy the Pythagorean theorem.

Pythagorean triples, especially primitive ones, are instrumental in exploring many mathematical concepts and have practical applications in various fields, such as architecture and engineering, where right-angled triangles frequently appear.