Question

the pythagorean theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse by the formula $a^{2}+b^{2}=c^{2}$. if $a$ is a rational number and $b$ is a rational number, why could $c$ be an irrational number? the left side of the equation will result in a rational number, which could be a non - perfect square. the left side of the equation will result in a rational number, which is a perfect square. the product of two rational numbers is rational, and the sum of two rational numbers is irrational.

Explanation:

If \(a\) and \(b\) are rational, \(a^{2}\) and \(b^{2}\) are rational, and their sum \(a^{2}+b^{2}\) is rational. But if \(a^{2}+b^{2}\) is a non - perfect square, then \(c=\sqrt{a^{2}+b^{2}}\) is irrational. The sum of two rational numbers is rational, not irrational. If \(a^{2}+b^{2}\) is a perfect square, \(c\) is rational.

Answer:

The left side of the equation will result in a rational number, which could be a non - perfect square.