Problem 169 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use the square root property to determine the length of a right triangle's side, I don't even bother to list the negative square root.

Step-by-Step Solution

Answer

The statement makes sense because when using the square root property to determine the length of a right triangle's side, only the positive square root is relevant. Distances, such as the sides of a triangle, cannot be negative.

Step 1: Understanding the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled 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. This can be written as: \(a^2 + b^2 = c^2\) where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

Step 2: Understanding the square root

In mathematics, the square root of a number x is a value that, when multiplied by itself, gives the original number x. The square root of x is usually denoted by \( \sqrt{x} \) or \( x^{1/2} \). However, two numbers meet this definition: the positive square root (or principal square root) and the negative square root. For any real number x > 0, if y is the positive square root of x, then -y is the negative square root of x. Hence, we usually have: \( \sqrt{x} = ± y \).

Step 3: Analyzing the claim

In the context of a right triangle, the sides (a, b, c) represent lengths or distances, which are always nonnegative in the real world. So, when we calculate the lengths of the sides using the square root property, we only consider the positive square root. The negative square root doesn't make sense in this context because length or distance can't be negative. This makes the claim plausible.