Answer:
Step-by-step explanation:
7. False. Opposite angles of a rhombus are congruent, not necessarily supplementary unless it's a square.
8. False. Parallelograms' consecutive angles must be supplementary. 168 and 22 do not add up to 180.
9. False. Rhombuses are not the only quadrilateral with perpendicular diagonals. Kites also have perpendicular diagonals.
10. Area of a regular polygon is:
A = 1/2 aP
where a is the apothem and P is the perimeter.
We're given a = 24.78, but we need to find the side length.
The interior angle of a regular polygon is:
θ = (n - 2) × 180° / n
So a hexagon with 6 sides has an interior angle of:
θ = (6 - 2) × 180° / 6
θ = 120°
If we draw lines from the bottom corner to the center, we get a 30-60-90 triangle. Therefore:
(s/2) × √3 = 24.78
s/2 = 14.307
s = 28.61
So the perimeter is:
P = 6s
P = 171.68
And the area is:
A = 1/2 aP
A = 1/2 (24.78) (171.68)
A = 2127.13 cm²
11. Area of a regular polygon is:
A = 1/2 aP
where a is the apothem and P is the perimeter.
We're given s = 11, but we need to find the apothem.
The interior angle of a regular polygon is:
θ = (n - 2) × 180° / n
So a polygon with 11 sides has an interior angle of:
θ = (11 - 2) × 180° / 11
θ = 1620/11 °
If we draw lines from the bottom corner to the center, we get a right triangle with a base angle of θ/2. Therefore:
tan (θ/2) = a / (s/2)
a = s/2 tan(θ/2)
a = 11/2 tan(810/11 °)
a = 18.73
The perimeter is:
P = 11s
P = 121
And the area is:
A = 1/2 aP
A = 1/2 (18.73) (121)
A = 1133.24 in²
