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?

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. Let's denote the height at points A and B as \( h_A \) and \( h_B \), respectively. The distance \( DC \) is given as \( 5\sqrt{3} \).

1. Using Triangle ADC:

   • In triangle ADC, angle \( ADC = 45^\circ \).
   • From the definition of the tangent function, we have: \[ \tan(45^\circ) = 1 = \frac{h_A}{DC} \]
   • Thus, \( h_A = DC \). Given \( DC = 5\sqrt{3} \): \[ h_A = 5\sqrt{3} \]

2. Using Triangle BDC:

   • In triangle BDC, angle \( BDC = 30^\circ \).
   • Similarly, using the tangent function, we have: \[ \tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{h_B}{DC} \]
   • Hence, \[ h_B = DC \cdot \tan(30^\circ) = 5\sqrt{3} \cdot \frac{1}{\sqrt{3}} = 5 \]

3. Finding the distance between points A and B:

   • The distance \( AB \) is simply the difference in heights of points A and B: \[ AB = h_A - h_B = 5\sqrt{3} - 5 \]

4. Calculating the value:

   • We know \( \sqrt{3} \approx 1.732 \): \[ 5\sqrt{3} \approx 5 \cdot 1.732 = 8.66 \]
   • Thus: \[ AB \approx 8.66 - 5 = 3.66 \]

Thus, the distance between points A and B on the flagpole is approximately \( 3.66 \) feet.

Thus, the answer is 3.66 feet.