19. A ladder that is 10 feet long is 6 feet from the base of the wall. How far up the wall does the ladder reach?

## Step 1<br />This problem involves the application of the Pythagorean theorem, which is a fundamental relation in Euclidean geometry among the three sides of a right triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be written as:<br />### \(a^2 + b^2 = c^2\)<br />where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides.<br /><br />## Step 2<br />In this problem, the ladder acts as the hypotenuse of the right triangle, with a length of 10 feet. The distance from the base of the ladder to the wall forms one side of the triangle, with a length of 6 feet. We are asked to find the height of the wall that the ladder reaches, which is the remaining side of the triangle.<br /><br />## Step 3<br />We can rearrange the Pythagorean theorem to solve for the unknown side (height of the wall):<br />### \(b = \sqrt{c^2 - a^2}\)<br /><br />## Step 4<br />Substituting the given values:<br />### \(b = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8\)<br /><br />Therefore, the ladder reaches 8 feet up the wall.