Question

The diagram shows two different nature trails in a state park. The solid line shows the Dogwood Trail. The dashed line shows the Elm Trail.

2 right triangles. One triangle has sides with lengths 5 kilometers, a, and hypotenuse 13 kilometers. The second triangle has sides with lengths 3 kilometers, b, and hypotenuse 5 kilometers.
Which of the following statements are true about the lengths of the trails? Check all that apply.
The total length of the Dogwood Trail is 16 kilometers.
The total length of the Dogwood Trail is 24 kilometers.
The Elm Trail is longer than the Dogwood Trail.
The difference between the lengths of the trails is 2 kilometers.
The difference between the lengths of the trails is 6 kilometers.

Answer 1

To analyze the lengths of the trails based on the information given in the problem, we first need to find the lengths of the Dogwood Trail and the Elm Trail using the properties of right triangles.

### For the Dogwood Trail:

We have a right triangle with two sides measuring 5 kilometers and \( a \) kilometers, and the hypotenuse is 13 kilometers. According to the Pythagorean theorem:

\[ a^2 + 5^2 = 13^2 \]
\[ a^2 + 25 = 169 \]
\[ a^2 = 169 - 25 \]
\[ a^2 = 144 \]
\[ a = \sqrt{144} = 12 \]

So the sides of the triangle are 5 km, 12 km, and 13 km.

The total length of the Dogwood Trail is:
\[ 5 + 12 + 13 = 30 \text{ kilometers} \]

### For the Elm Trail:

We have a right triangle with sides measuring 3 kilometers and \( b \) kilometers, and the hypotenuse is 5 kilometers. According to the Pythagorean theorem:

\[ b^2 + 3^2 = 5^2 \]
\[ b^2 + 9 = 25 \]
\[ b^2 = 25 - 9 \]
\[ b^2 = 16 \]
\[ b = \sqrt{16} = 4 \]

So the sides of the triangle are 3 km, 4 km, and 5 km.

The total length of the Elm Trail is:
\[ 3 + 4 + 5 = 12 \text{ kilometers} \]

### Now, evaluating the provided statements:

1. **The total length of the Dogwood Trail is 16 kilometers.**
**False** (it is 30 kilometers)

2. **The total length of the Dogwood Trail is 24 kilometers.**
**False** (it is 30 kilometers)

3. **The Elm Trail is longer than the Dogwood Trail.**
**False** (Dogwood Trail is 30 km, Elm Trail is 12 km)

4. **The difference between the lengths of the trails is 2 kilometers.**
**False** (The difference is \( 30 - 12 = 18 \) kilometers)

5. **The difference between the lengths of the trails is 6 kilometers.**
**False** (the difference is \( 30 - 12 = 18 \) kilometers)

### Conclusion:

All the statements provided are **false** based on the calculations of the lengths of the Dogwood and Elm Trails.