Chapter 5, Problem 3GA

Estimated Time: 25–30 minutes

Group Size: 2

Right triangles occur in many applications of mathematics. By definition, a right triangle is a triangle that contains a 90° angle. The two shorter sides in a right triangle are referred to as the “legs,” and the longest side is called the “hypotenuse.” In the triangle shown, the legs are labeled as a and b, and the hypotenuse is labeled as c.

Right triangles have an important property that the sum of the squares of the two legs of a right triangles have an important property that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse. This fact is referred to as the Pythagorean theorem. In symbols, the Pythagorean theorem is stated as:

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

Now equate the two expressions representing the area of the large outer square:

To determine

To calculate: The expression representing the area of large outer square. The figure is as follows:

Answer to Problem 3GA

Solution:

Equating the area of large outer square to the sum of area of Inner Square and area of four right triangle gives $\underset{\_}{a^2+b^2=c^2}$.

Explanation of Solution

Given Information:

The area of Inner Square is ${\left(c\right)}^2$, area of large outer square is ${\left(a+b\right)}^2$ and, area of four right triangles is $4\cdot \left(\frac{1}{2}ab\right)$. The figure is as follows:

Formula Used:

The area of square is ${\left(\text{side}\right)}^2$.

The area of right-angle triangle is $\frac{1}{2}\times \text{base}\times \text{height}$.

Calculation:

Consider the given figure.

Steps to determine the area of large outer square:

Step1: First determine the area of Inner Square and area of four right triangles.

Step2: Then add both areas which is equal to area of large outer square.

Area of Inner Square is $c^2$ and, area of four right triangles is $4\cdot \left(\frac{1}{2}ab\right)$.

Now, add these two areas and equate it to the area of large outer square.

${\left(a+b\right)}^2=c^2+4\cdot \left(\frac{1}{2}ab\right)$

Clear the parenthesis on both sides of the equation.

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

Subtract $2ab$ from both sides,

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

Thus, equating the area of large outer square to the sum of area of Inner Square and area of four right triangle gives $a^2+b^2=c^2$.