Question

A triangle can be formed by drawing line segments on a map of Texas connecting the cities of​ Dallas, Houston, and San Antonio​ (see figure).
Dallas
Houston
San Antonio
3 in
3.75 in
4.125 in
Question content area bottom
Part 1
If the actual distance from San Antonio to Houston is approximately 194 ​miles, use the lengths of the line segments indicated in the figure along with similar triangles to approximate
a. the actual distance from Dallas to Houston.
b. the actual distance from Dallas to San Antonio.

Answer 1

To use the lengths of the line segments on the map and the actual distance from San Antonio to Houston to approximate the actual distances from Dallas to Houston and from Dallas to San Antonio, we can set up a proportion based on similar triangles.

**Step 1: Identify the lengths of the segments on the map.**

- San Antonio to Houston: \(3.75 \: \text{inches}\)
- Dallas to Houston: \(3 \: \text{inches}\)
- Dallas to San Antonio: \(4.125 \: \text{inches}\)

**Step 2: Use the distance from San Antonio to Houston to set up proportions.**

Given that the actual distance from San Antonio to Houston is approximately \(194\) miles, we can set up the proportion for Dallas to Houston and Dallas to San Antonio.

### Part a: Actual distance from Dallas to Houston

1. **Setting up the proportion:**

\[
\frac{\text{Dallas to Houston (inches)}}{\text{San Antonio to Houston (inches)}} = \frac{\text{Dallas to Houston (miles)}}{\text{San Antonio to Houston (miles)}}
\]

Plugging in the values we have:

\[
\frac{3 \: \text{in}}{3.75 \: \text{in}} = \frac{d}{194 \: \text{miles}}
\]

where \(d\) is the actual distance from Dallas to Houston.

2. **Cross multiplying to solve for \(d\):**

\[
3d = 3.75 \times 194
\]

\[
3d = 727.5
\]

\[
d = \frac{727.5}{3}
\]

\[
d = 242.5 \: \text{miles}
\]

**Thus, the actual distance from Dallas to Houston is approximately \(242.5\) miles.**

### Part b: Actual distance from Dallas to San Antonio

1. **Setting up the proportion for Dallas to San Antonio:**

\[
\frac{\text{Dallas to San Antonio (inches)}}{\text{San Antonio to Houston (inches)}} = \frac{\text{Dallas to San Antonio (miles)}}{\text{San Antonio to Houston (miles)}}
\]

Plugging in the values:

\[
\frac{4.125 \: \text{in}}{3.75 \: \text{in}} = \frac{d}{194 \: \text{miles}}
\]

2. **Cross multiplying to solve for \(d\):**

\[
4.125 \times 194 = 3.75d
\]

Calculating \(4.125 \times 194\):

\[
4.125 \times 194 = 799.5
\]

Thus,

\[
799.5 = 3.75d
\]

\[
d = \frac{799.5}{3.75}
\]

Calculating \(d\):

\[
d \approx 213.2 \: \text{miles}
\]

**Thus, the actual distance from Dallas to San Antonio is approximately \(213.2\) miles.**

### Summary
- Actual distance from Dallas to Houston: **242.5 miles**
- Actual distance from Dallas to San Antonio: **213.2 miles**