Answer:
1) The area of the regular hexagon is approximately 11.256 unit²
2) The coordinates of the center is (2, 3)
3) The base length is approximately 2.08 units
4) The height of the hexagon is approximately 3.61 units
Step-by-step explanation:
1) The length, lₐ, of the apothem is given as follows;
[tex]l_a = \sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}[/tex]
Therefore, for the apothem having coordinates, (2, 3), and (3, 1.5), we have;
[tex]l_a = \sqrt{\left (3-1.5 \right )^{2}+\left (2-3 \right )^{2}} = \sqrt{1.5^2 + (-1)^2} =\sqrt{3.25}[/tex]
The length of half half of one side, S/2 = a × tan(30°) = √(3.25)/√3
The length of the base = 2 × √(3.25)/√3 ≈ 2.082 units
The perimeter, P = 6 × 2× √(3.25)/√3
The area, A = 1/2 × P × a = 1/2 × 6 × 2 × √(3.25)/√3 × √3.25 = (13·√3)/2
A = (13·√3)/2 unit²
The area of the regular hexagon, A = (13·√3)/2 unit² ≈ 11.256 unit²
2) The coordinates of the center = (2, 3)
3) The base, 'b', length by Pythagorean theorem is given as follows;
b = √(a² + (S/2)²) = √((√(3.25))² + (√(3.25)/√3)²) = √(3.25 + 3.25/3) = √(13/3) = (√39)/3
The base length, b = (√39)/3 units ≈ 2.08 units
4) The height of the hexagon, h = 2 × The length of the apothem, lₐ
The length, lₐ, of the apothem is given as follows;
[tex]l_a = \sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}[/tex]
Given the apothem coordinates, (2, 3), and (3, 1.5), we have;
[tex]l_a = \sqrt{\left (3-1.5 \right )^{2}+\left (2-3 \right )^{2}} = \sqrt{1.5^2 + (-1)^2} =\sqrt{3.25}[/tex]
The height of the hexagon, h = 2 × √3.25 units ≈ 3.61 units.