An artist is using a coordinate plane to plan a string design for a wall. The artist plans to hammer a nail at each vetex of regular hexagon. then the artist will string to connect the vertices to make the hexagon shape. the artiststarts with a circle centered at the origin and places first vertex at (8,0) where are the other vertices? give coordinates
The first vertex is at 0° with the x-axis. The other vertices are at respectively θ= 60, 120, 180, 240 and 300° with the x-axis.
The corresponding coordinates are at:
( 8cos(θ), 8sin(θ) )
Example:
At θ=0° the vertex is
(8cos(0°), 8sin(0°))
=(8,0)
To determine the coordinates of the other vertices of the regular hexagon, we need to understand the pattern of the hexagon's shape.
A regular hexagon has six equal sides and six equal angles. Let's assume that the first vertex is located at (8,0), which means it is 8 units to the right of the origin.
Starting from this vertex, we can use trigonometry to calculate the coordinates of the remaining vertices. The angle between each adjacent vertex is 60 degrees in a regular hexagon.
Step 1: Calculate the angle in radians:
The angle in radians can be calculated by converting 60 degrees to radians using the formula: radians = degrees * (π / 180).
Plugging in the value, we get: radians = 60 * (π / 180) = π / 3 (approximately 1.047).
Step 2: Determine the coordinates of the remaining vertices:
To determine the coordinates of the remaining vertices, we will rotate the initial vertex (8,0) counterclockwise by multiples of 60 degrees.
The coordinates of the second vertex can be found by applying a rotation to the first vertex:
x = 8 * cos(1 * (π / 3)) = 8 * cos(π / 3) ≈ 4
y = 8 * sin(1 * (π / 3)) = 8 * sin(π / 3) ≈ 6.93
So, the coordinates of the second vertex are approximately (4, 6.93).
To find the coordinates of the other vertices, repeat the process by multiplying the angle (in radians) by 2, 3, 4, 5, and 6.
The coordinates of the remaining vertices are approximately:
Vertex 3: (-4, 6.93)
Vertex 4: (-8, 0)
Vertex 5: (-4, -6.93)
Vertex 6: (4, -6.93)
Therefore, the coordinates of the six vertices of the regular hexagon, starting from the initial vertex at (8,0), are approximately:
(8, 0), (4, 6.93), (-4, 6.93), (-8, 0), (-4, -6.93), (4, -6.93).