Question

Calculate the side measures (Miami, FL to Bermuda, Bermuda to San Juan, Puerto Rico, and San Juan, Puerto Rico to Miami, FL). They can apply the Triangle Inequality Theorem to confirm the sum of two sides is greater than the measure of the third side of the triangle. Students can find the measure of the angles of the triangle using a protractor or technology, and apply the Triangle Sum Theorem to confirmthat the angles sum tp 180 degrees.

Answer 1

To calculate the side measures of a triangle formed by the points Miami, FL; Bermuda; and San Juan, Puerto Rico, we first need to find the distances between each pair of locations. Here are the approximate distances:

1. **Miami, FL to Bermuda**: Approximately 1,045 miles.
2. **Bermuda to San Juan, Puerto Rico**: Approximately 743 miles.
3. **San Juan, Puerto Rico to Miami, FL**: Approximately 1,018 miles.

Now, we have the side measures:

- **AB (Miami to Bermuda)**: 1,045 miles
- **BC (Bermuda to San Juan)**: 743 miles
- **CA (San Juan to Miami)**: 1,018 miles

### Applying the Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. We can check this for our triangle:

1. **AB + BC > CA**
- 1,045 + 743 > 1,018
- 1,788 > 1,018 (True)

2. **AB + CA > BC**
- 1,045 + 1,018 > 743
- 2,063 > 743 (True)

3. **BC + CA > AB**
- 743 + 1,018 > 1,045
- 1,761 > 1,045 (True)

Since all three conditions of the Triangle Inequality Theorem are satisfied, these points can indeed form a triangle.

### Finding the Angles

To find the measures of the angles, we can apply the Law of Cosines:

1. **Angle A (Miami, FL):**
\[
\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}
\]
Where:
- \(a = BC = 743\)
- \(b = CA = 1,018\)
- \(c = AB = 1,045\)

Plugging in the values:
\[
\cos(A) = \frac{(1,018^2) + (1,045^2) - (743^2)}{2 \cdot 1,018 \cdot 1,045}
\]

Calculate:
\[
1,018^2 \approx 1,036,324,\quad 1,045^2 \approx 1,092,025,\quad 743^2 \approx 552,649
\]

Substituting:
\[
\cos(A) = \frac{(1,036,324) + (1,092,025) - (552,649)}{2 \cdot 1,018 \cdot 1,045} \approx \frac{1,575,700}{2,128,580} \approx 0.741
\]

Thus, \(A \approx \cos^{-1}(0.741) \approx 42.60^\circ\).

2. **Angle B (Bermuda):**
\[
\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}
\]
\[
\cos(B) = \frac{(743^2) + (1,045^2) - (1,018^2)}{2 \cdot 743 \cdot 1,045}
\]
Substituting for calculations:
\[
\cos(B) = \frac{(552,649) + (1,092,025) - (1,036,324)}{2 \cdot 743 \cdot 1,045} \approx \frac{608,350}{1,550,110} \approx 0.392
\]

Thus, \(B \approx \cos^{-1}(0.392) \approx 66.37^\circ\).

3. **Angle C (San Juan, Puerto Rico)**:
To find angle C, you can use the Triangle Sum Theorem:
\[
C = 180^\circ - A - B
\]
Substituting:
\[
C = 180^\circ - 42.60^\circ - 66.37^\circ \approx 71.03^\circ
\]

### Summary

The side lengths are:
- Miami to Bermuda: **1,045 miles**
- Bermuda to San Juan: **743 miles**
- San Juan to Miami: **1,018 miles**

The angle measures for the triangle formed by these three points are approximately:
- Angle A (Miami, FL): **42.60°**
- Angle B (Bermuda): **66.37°**
- Angle C (San Juan, Puerto Rico): **71.03°**

Verifying, \(A + B + C \approx 42.60° + 66.37° + 71.03° \approx 180°\).

Thus, the calculations for both the sides and angles satisfy the properties of a triangle.