A regular octagon is inscribed in a circle with a radius of 10 cm. What is the length of one side of the octagon?
2 answers:
Answer:
The length of one side of the octagon is 7.65 cm
Step-by-step explanation:
The parameters given are;
A regular octagon inscribed in a circle of radius, r, of 10 cm.
The length of each side is found from the isosceles triangle formed by the radius and one side of the octagon
The sum of interior angles in a polygon, ∑θ = 180 × (n - 2)
Where;
n = The number of sides of the polygon
θ = The interior angle of the polygon
For the octagon, we have;
n = 8, therefore;
∑θ = 180 × (8 - 2) = 1080
Given that there are eight equal angles in a regular octagon, we have;
∑θ = 8 × θ
= 1080
θ = 1080/8 = 135°
The sum of angles at the center of the circle = 360
Therefore, the angle at the center (tip angle) of the isosceles triangle formed by the radius and one side of the octagon = 360/8 = 45°
The base angles of the isosceles triangle is therefore, (180 - 45)/2 = 67.5° = θ/2
The length of the base of the isosceles triangle formed by the radius and one side of the octagon = The length of one side of the octagon
From trigonometric ratios, the length of the base of the isosceles triangle is therefore;
2 × r × cos(θ/2) = 2×10 × cos(67.5°) = 7.65 cm
The length of the base of the isosceles triangle = 7.65 cm = The length of one side of the octagon.
You might be interested in
<em>Equation b) (-3 + 5i) (1) = -3 + 5i demonstrates the multiplicative identity property </em>
<h3>Further explanation</h3>There are several properties in integer multiplication operations
- 1. closed property
Multiplication between integers will produce integers too
- 2. commutative property
a x b = b x a
- 3. associative property
ax (bxc) = (axb) xc
- 4. identity
ax1 = 1 x a = a
- 5. distributive property
* addition
ax (b + c) = axb + axc
* substraction
ax (b-c) = axb - axc
The Multiplicative Identity is <u><em>1 </em></u>
<u><em>The multiplicative identity property states that any number multiplied by 1 will produce an original number. </em></u>
We complete the answer choices available from the problem above:
a) (- 3 + 5i) + 0 = -3 + 5i
O is an identity in the sum operation, the statement is false
b) (-3 + 5i) (1) = -3 + 5i
1 is a Multiplicative Identity of multiplication, so the statement is true
c) (-3 + 5i) (-3 + 5i) = -16-30i
d) (-3 + 5i) (-3 + 5i) = 16 + 30i
choice c and is a multiplication factor, so the statement is false
<h3>Learn more </h3>
the substitution method
The work of a student to solve a set of equations
Keywords: Multiplicative Identity property, integers
