Question

Two ropes, AD and BD, are tied to a peg on the ground at point D. The other ends of the ropes are tied to points A and B on a flagpole, as shown below:

Two ropes, AD and BD, are tied to a peg on the ground at point D. The other ends of the ropes are tied to points A and B on a flagpole. Angle ADC measures 45 degrees, and angle BDC measures 30 degrees. The length of DC is 5 multiplied by square root of 3.

Angle ADC measures 45°, and angle BDC measures 30°. What is the distance between the points A and B on the flagpole? (1 point)

3.66 feet

13.66 feet

8.66 feet

5 feet

Answer 1

To find the distance between points A and B on the flagpole, we can use trigonometry based on the given angles and the length of DC.

1. **Identify the lengths of AD and BD using the tangent function:**
- For triangle ADC:
\[
\tan(45^\circ) = \frac{AC}{DC}
\]
Since \(\tan(45^\circ) = 1\),
\[
AC = DC = 5\sqrt{3} \, \text{feet}
\]

- For triangle BDC:
\[
\tan(30^\circ) = \frac{BC}{DC}
\]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\),
\[
BC = DC \times \tan(30^\circ) = 5\sqrt{3} \times \frac{1}{\sqrt{3}} = 5 \, \text{feet}
\]

2. **Find the total distance between points A and B on the flagpole:**
\[
AB = AC + BC = 5\sqrt{3} + 5
\]

3. **Calculate \(AB\):**
Using \(\sqrt{3} \approx 1.732\),
\[
5\sqrt{3} \approx 5 \times 1.732 = 8.66 \, \text{feet}
\]

4. **Thus, the total distance is:**
\[
AB \approx 8.66 + 5 = 13.66 \, \text{feet}
\]

Therefore, the distance between points A and B on the flagpole is **13.66 feet**.