20- ft ladder is standing up against a wall. The distance between the base of the ladder and the wall is 4 ft tess than the distance between the top of the ladder nd the base of the wall. Find the distance between the base of the ladder and the wall.

To solve this, we can use the Pythagorean theorem. Here, the ladder forms a right triangle with the wall and the ground. Let \( x \) be the distance between the base of the ladder and the wall. According to the problem, the distance from the top of the ladder to the base of the wall is \( x + 4 \). Using the Pythagorean theorem: \[ x^2 + (x + 4)^2 = 20^2 \] Expanding the equation: \[ x^2 + (x^2 + 8x + 16) = 400 \] \[ 2x^2 + 8x + 16 = 400 \] \[ 2x^2 + 8x - 384 = 0 \] Dividing the entire equation by 2 simplifies to: \[ x^2 + 4x - 192 = 0 \] Now, to find \( x \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = 4, c = -192 \). Calculating the discriminant: \[ b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-192) = 16 + 768 = 784 \] So the solutions are: \[ x = \frac{-4 \pm 28}{2} \] This gives us two potential solutions: \[ x = \frac{24}{2} = 12 \quad \text{and} \quad x = \frac{-32}{2} = -16 \] Since the distance cannot be negative, the distance between the base of the ladder and the wall is \( 12 \) feet. --- Did you know that as far back as the ancient Greeks, mathematicians like Pythagoras were already studying relationships in right triangles? The Pythagorean theorem, discovered by them, has become a cornerstone in geometry, helping solve numerous practical problems involving heights and distances—even in construction design today! As fun as it is to solve these problems, remember that it's easy to mix up which side of the equation sides correspond to the ladder, wall, and ground! Always double-check which distances you are using—getting the relationship wrong can lead to incorrect conclusions, so draw a quick sketch if needed!