Question

One square has a side length of s. Another square has a side length of t. a) Write an algebraic expression that describes the sum of the areas of the two squares. b) Evaluate your expression for various whole number values of s and t. c) Does the resulting sum ever represent the area of a single square with a whole-number side length? Explain.

Answer 1

The answer will be on the explanation side!

Explanation:

a) The sum of the areas of the two squares can be expressed as:

s^2 + t^2

b) Here are some examples of evaluating the expression for various whole number values of s and t:

- If s = 2 and t = 3, then the sum of the areas is 2^2 + 3^2 = 4 + 9 = 13.

- If s = 5 and t = 5, then the sum of the areas is 5^2 + 5^2 = 25 + 25 = 50.

- If s = 0 and t = 1, then the sum of the areas is 0^2 + 1^2 = 1.

c) In general, the resulting sum does not represent the area of a single square with a whole-number side length, unless s and t happen to satisfy a certain condition. In order for the sum to represent the area of a single square with a whole-number side length, s^2 + t^2 must be a perfect square.

For example, if s = 3 and t = 4, then the sum of the areas is 3^2 + 4^2 = 9 + 16 = 25, which is a perfect square (5^2). So in this case, the resulting sum represents the area of a single square with a whole-number side length.

However, in general, there are infinitely many pairs of whole numbers s and t for which s^2 + t^2 is not a perfect square, so the resulting sum does not represent the area of a single square with a whole-number side length.