Sarah is building a dog house. She has created the rectangular house and just needs to build the roof to fit. She designs a triangle-shaped roof by using two sheets of wood, forming a 90° angle at the peak. Looking at the front of the dog house, the distance from either wall is 5 feet across. If the triangle portion is symmetrical, what is the distance from the top of the wall portion to the highest part of the triangle peak? Round your answer to the nearest tenth.(1 point)

Answer 1

To find the distance from the top of the wall portion to the highest part of the triangle peak, we can analyze the triangle formed by the roof.

Since the triangle is symmetrical and forms a right triangle with a 90° angle at the peak, the distance from either wall to the peak is half of the total distance across. Given that the distance from either wall is 5 feet, the total distance across is:

\[ 5 \text{ feet} + 5 \text{ feet} = 10 \text{ feet} \]

Since the triangle forms a right triangle, the two sides leading up to the peak will be equal. We can visualize the triangle like this:

1. The base is 10 feet wide (total distance across).
2. The height of the triangle (distance from the top of the wall to the peak) is the unknown we wish to find.

By using the properties of a right triangle, we know that the base of each right triangle formed by the peak and walls measures 5 feet (half of the total distance). Let \( h \) be the height from the top of the walls to the peak.

Using the Pythagorean theorem, we can express the relationship as:

\[ \text{Base}^2 + \text{Height}^2 = \text{Hypotenuse}^2 \]

In this scenario, if we assume the length of each side (slope of the roof) is the same, but we only need to find the height, we can choose a reference right triangle along the center point (the peak).

We assume the two legs of the triangle are equal since it is symmetrical:

Using \( h \) as the height from the wall to the peak, \[ (5)^2 + h^2 = (\text{Hypotenuse})^2 \]

However, in order to find the simplest representation without the length of the hypotenuse given, we recognize that the height we want simply follows from geometry principles, which establish this construction:

With a symmetrical triangle: If:

• Base \( b = 10 \) feet (the width from wall to wall),
• The left wall to peak: \( 5 \) feet.

The height \( h \) will depend upon the triangle's proportions.

To find either vector involving slope not required here, with equal inclination, you respond directly along \( h \):

\[ h = 5 \text{ feet} \]

Thus the height from the top of the wall to the peak of the triangle is 5 feet.

The final answer is:

\[ \boxed{5.0} \text{ feet } \]

This indicates the vertical clearance to the top point as typically rounded accordingly to the specification required.