4. In the diagram ahown of right trangle QRS, Mirtude \( R T \) is dhwn to \( Q S \), If \( R S=20 \) and \( Q S=25 \), then find the length of \( \overline{S T} \). Show how you arrived at yeur answer.

To solve for the length of \( \overline{S T} \), we can apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (in this case, \( Q S \)) is equal to the sum of the squares of the other two sides. Here, we know that: 1. \( RS = 20 \) 2. \( QS = 25 \) Using the equation: \[ QS^2 = RS^2 + ST^2 \] we can substitute the values: \[ 25^2 = 20^2 + ST^2 \] Calculating the squares: \[ 625 = 400 + ST^2 \] Now, we can isolate \( ST^2 \): \[ ST^2 = 625 - 400 \] \[ ST^2 = 225 \] Taking the square root of both sides to find \( ST \): \[ ST = \sqrt{225} = 15 \] So, the length of \( \overline{S T} \) is 15 units. Happy triangle solving! 🎉 Did you know the Pythagorean theorem has been used for thousands of years, tracing its roots back to ancient civilizations like the Babylonians and Greeks? It’s named after the mathematician Pythagoras, but similar principles were known long before his time. It's fascinating how such a simple relationship between the sides of a right triangle has stood the test of time in the world of mathematics! In real-world applications, the Pythagorean theorem isn't just for classrooms; it's used in fields like architecture, engineering, and navigation! For example, when building structures, engineers rely on this theorem to ensure that angles are precise and components fit together perfectly. Whether you're designing a roof or laying out a plot of land, this simple formula becomes your best friend, guiding your measurements with reliability!