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:

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. 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.