Question

question 4 of 10

a 14-foot ladder is leaning against a building. the ladder makes a 45-degree angle with the building. how far up the building does the ladder reach?

a. \\(7\sqrt{2}\\) feet
b. \\(28\sqrt{2}\\) feet
c. \\(14\sqrt{2}\\) feet
d. 7 feet

Explanation:

Model the physical scenario

We model the ladder, ground, and building as a right triangle.

• The ladder is the hypotenuse: \(c = 14\text{ feet}\).
• The angle between the ladder and the building is \(45^\circ\).
• The building and ground meet at a \(90^\circ\) angle.
• This forms a \(45^\circ\)-\(45^\circ\)-\(90^\circ\) right triangle.

Apply the triangle properties

Using the 45-45-90 Triangle Theorem and Special Right Triangles knowledge points:

$$ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} $$

$$ 14 = x\sqrt{2} $$

Solve for the height

Using the Special Right Triangles knowledge point:

$$ x = \frac{14}{\sqrt{2}} = \frac{14\sqrt{2}}{2} = 7\sqrt{2}\text{ feet} $$

Answer:

• A. \(7\sqrt{2}\) feet (Correct answer)
• B. \(28\sqrt{2}\) feet
• C. \(14\sqrt{2}\) feet
• D. 7 feet