Question

a 15 foot ladder is leaning against a house. how far is the top of the ladder from the ground if the ladder is 12 feet away from the house

Explanation:

Step1: Identify the problem type

This is a right triangle problem where the ladder is the hypotenuse (\(c = 15\) feet), the distance from the house is one leg (\(a = 12\) feet), and we need to find the other leg (\(b\)) which is the height from the ground to the top of the ladder. We use the Pythagorean theorem: \(a^{2}+b^{2}=c^{2}\).

Step2: Rearrange the formula to solve for \(b\)

We rearrange the Pythagorean theorem to \(b=\sqrt{c^{2}-a^{2}}\).

Step3: Substitute the values

Substitute \(c = 15\) and \(a = 12\) into the formula: \(b=\sqrt{15^{2}-12^{2}}\).

Step4: Calculate the squares

Calculate \(15^{2}=225\) and \(12^{2}=144\). So the formula becomes \(b=\sqrt{225 - 144}\).

Step5: Subtract the values inside the square root

\(225-144 = 81\), so \(b=\sqrt{81}\).

Step6: Calculate the square root

\(\sqrt{81}=9\).

Answer:

The top of the ladder is 9 feet from the ground.