Question

a ladder is leaning against a building. the ladder makes a 45 degree angle with the building and reaches 10 feet up the side of the building. how long is the ladder? 20\sqrt{2} feet 10\sqrt{2} feet 10 feet 5\sqrt{3} feet

Explanation:

Step1: Identify the triangle type

The triangle formed by the building, the ground, and the ladder is a right - angled triangle with one angle equal to \(45^{\circ}\). Since the sum of angles in a triangle is \(180^{\circ}\) and one angle is \(90^{\circ}\), the other non - right angle is also \(45^{\circ}\) (because \(180-(90 + 45)=45\)). So, it is a \(45 - 45-90\) triangle.

Step2: Recall the side - length ratio of a \(45 - 45-90\) triangle

In a \(45 - 45-90\) triangle, if the length of each of the legs (the sides adjacent to the \(45^{\circ}\) angles) is \(a\), the length of the hypotenuse \(c\) is given by \(c = a\sqrt{2}\). Here, the legs of the right - angled triangle (the height on the building and the distance from the base of the building to the base of the ladder) are \(a = 10\) feet.

Step3: Calculate the length of the ladder (hypotenuse)

Using the formula \(c=a\sqrt{2}\) with \(a = 10\), we get \(c=10\sqrt{2}\) feet.

Answer:

\(10\sqrt{2}\) feet