If the length of each side of a regular hexagon is 10 ft, the what is the distance from a vertex to the center?

Andrea and Carlos left the airport at the same time. Andrea flew at 180 mph on a course with bearing 80 degrees, and Carlos flew at 240 mph on a course with bearing 210 degrees. How far apart were they after 3 hr? Round the nearest 10th.

1st one:

Too easy. Joining all vertices of a hexagon to the centre creates 6 equilateral triangles.
So if the side of the hexagon is 10 ft, all sides must be 10 ft. so the distance to the centre is 10 ft

2nd:
you have a triangle with sides 540 and 720 with an angle of 160° between them
let the distance between them be x
by cosine law:
x^2 = 540^2+720^2-2(540)(720)cos160°
= .....
you do the button-pushing.

To find the distance from a vertex to the center of a regular hexagon, we can make use of the property that the radius of a regular hexagon is equal to the length of each side divided by 2.

Given that the length of each side of the regular hexagon is 10 ft, we can calculate the radius by dividing the length of each side by 2:

Radius = 10 ft / 2 = 5 ft.

Therefore, the distance from a vertex to the center of the regular hexagon is 5 ft.

To find the distance between Andrea and Carlos after 3 hours, we can use the concept of relative motion.

Using the information provided, we know that Andrea has been flying at 180 mph on a course with a bearing of 80 degrees, and Carlos has been flying at 240 mph on a course with a bearing of 210 degrees.

To determine their relative motion, we need to consider the angle between their courses. The angle between their courses is the difference between their bearings: 210 degrees - 80 degrees = 130 degrees.

To find the distance between them after 3 hours, we can use the law of cosines:

Distance^2 = (Andrea's speed)^2 + (Carlos' speed)^2 - 2 * (Andrea's speed) * (Carlos' speed) * cos(angle between their courses).

Plugging in the values:

Distance^2 = (180 mph)^2 + (240 mph)^2 - 2 * (180 mph) * (240 mph) * cos(130 degrees).

Calculating:

Distance^2 = 32400 + 57600 - 2 * 180 * 240 * cos(130 degrees).

Distance^2 = 32400 + 57600 - 2 * 180 * 240 * (-0.6428).

Distance^2 = 32400 + 57600 + 69792.

Distance^2 = 159792.

Distance = √159792.

Distance ≈ 399.74 miles (rounded to the nearest 10th).

Therefore, after 3 hours, Andrea and Carlos are approximately 399.74 miles apart.