Question

Write an equation that relates the quantities. The square of the length of the hypotenuse \(c\) of a right triangle varies jointly with the sum of the squares of the lengths of its legs \(a\) and \(b .\) The constant of proportionality is 1

Short answer

The equation is c^2 = a^2 + b^2 .

Step-by-step solution

Identify Key Information

The problem states that there is a relationship between the square of the hypotenuse ( c ) and the sum of the squares of the legs ( a and b ) of a right triangle, with a constant of proportionality of 1. This hints at the Pythagorean theorem.

Understand the Proportional Relationship

Since the relationship is given to be a joint variation with a constant of proportionality of 1, this means that c^2 is directly proportional to a^2 + b^2 .

Set Up the Proportional Equation

Using the constant of proportionality (which is 1), write the equation as: c^2 = a^2 + b^2 .

Verify the Equation

Check that the equation c^2 = a^2 + b^2 correctly expresses the relationship where the square of the hypotenuse is equal to the sum of the squares of the legs, consistent with the Pythagorean theorem.

Key concepts

right triangle

A right triangle is a special kind of triangle that has one angle measuring exactly 90 degrees. This is called the right angle, and it is usually marked with a small square in triangle diagrams.
The sides of a right triangle have specific names: the two shorter sides are called 'legs,' and the longest side, which is opposite the right angle, is called the 'hypotenuse.' This unique triangle shape allows us to use the Pythagorean theorem to relate the lengths of the sides.

hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. In any right triangle, the hypotenuse can be found using the Pythagorean theorem. This states that the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (the legs, often labeled \(a\) and \(b\)).
Mathematically, this relationship is written as: \[ c^2 = a^2 + b^2 \].
Understanding this equation helps in solving many geometric problems involving right triangles.

joint variation

Joint variation occurs when a quantity varies directly with the product or sum of two or more other quantities. For the given exercise, it means that the square of the hypotenuse \(c^2\) varies jointly with the sum of the squares of the legs \(a^2\) and \(b^2\).
When we say 'varies jointly with' and the 'constant of proportionality is 1,' we mean that there is a direct and consistent relationship between these quantities. In simpler terms, the change in one quantity is directly related to changes in the other quantities by a set factor.

proportional relationship

A proportional relationship means that two quantities maintain a constant ratio or rate. For the Pythagorean theorem, this proportional relationship is crucial.
In the case of a right triangle, the square of the hypotenuse (\(c^2\)) is always proportional to the sum of the squares of the legs (\(a^2 + b^2\)). Given that the constant of proportionality is 1, our equation becomes:

• \( c^2 = a^2 + b^2 \)

This means that if you know any two sides of a right triangle, you can always find the third side using this consistent ratio.