T 5 - Mathematical Knowledge If a right triangle has one side that measures 5 cm and a hypotenuse that measures \( 5 \sqrt{2} \mathrm{~cm} \), what is the measure of the other side? A 5 cm B 25 cm C \( 5 \sqrt{3} \mathrm{~cm} \) D \( 10 \sqrt{2} \mathrm{~cm} \) Click the button or type the letter to the left of your answer.

To find the measure of the other side of the right triangle, we can use the Pythagorean theorem, which states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\): \[ a^2 + b^2 = c^2 \] Given one side \(a = 5\) cm and hypotenuse \(c = 5\sqrt{2}\) cm, we can rearrange and solve for \(b\): 1. Calculate \(c^2\): \[ c^2 = (5\sqrt{2})^2 = 25 \cdot 2 = 50 \] 2. Substitute into the equation: \[ 5^2 + b^2 = 50 \] \[ 25 + b^2 = 50 \] 3. Solve for \(b^2\): \[ b^2 = 50 - 25 = 25 \] 4. Take the square root: \[ b = \sqrt{25} = 5 \text{ cm} \] The measure of the other side is \(5\) cm, so the answer is **A**. Now, let’s dive into some interesting tidbits! The Pythagorean theorem isn't just a theory; it was known even in ancient civilizations like the Babylonians! They had a method for generating Pythagorean triples (like \(3, 4, 5\) and \(5, 12, 13\)) long before Pythagoras was born! Who knew math could have all this history behind it? If you ever find yourself confused with the Pythagorean theorem, just remember to always start with identifying which sides of the triangle you have. A common mistake is mixing up the hypotenuse and one of the legs. Always double-check your variables and keep your triangles clear – it can save you from math headaches later on!