Suppose a ladder 20 feet long is placed against a vertical wall 20 feet high . how far would the top of the ladder move down the wall by pulling out the bottom of the ladder 5 feet

The ladder leaning against the wall forms a right angled triangle with the gound and the wall. So we can use the formula:a² + b² = c²The ladder is the hypotenuse c²The vertical leg is b²The base or horizontal leg is a²We need to find the length of the base a²,  so:a² = c² - b²a² = 5² - 4²a² = 25 - 16a² = 9a = √9a = 3Therefore the bottom of the ladder must be 3 feet  from the wall.

Example:
If the 24 foot ladder is leaned on the house with the bottom 8 feet  from the base  the wall of the house, then it will form a right angled triangle.The base of the triangle will be 8 feet while the 24 foot ladder forms the hypotenuse. We need to find out the height from the base of that wall to the point the ladder touches the wall.Let the hypotenuse be CLet the base be BLet the height be AWe use this formula:   A squared + B squared = C squaredA sq + 8 sq = 24 sqA sq + 64 = 576A sq = 576 - 64A sq = 512A = Square root 512A = 22.6To the nearest 10th this will be 23So the answer is 23 feet.