Question

Trigonometry, due in a few hours, 100 pts

Answer 1

∠ABC = 120°

ΔABC = 16√3 cm²

Explanation:

Part (a)

Sum of interior angles of a regular polygon = (n - 2) × 180°
(where n is the number of sides)

⇒ Sum of interior angles of a regular hexagon = (6 - 2) × 180° = 720°

All the interior angles in a regular polygon are equal.

⇒ Interior angle = sum of interior angles ÷ number of sides

⇒ ∠ABC = 720° ÷ 6

⇒ ∠ABC = 120°

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Part (b)

Use the sine rule for area of a triangle:

`\frac12ab \sin(C)`

(where a and b are the sides and C is the included angle)

Given:

• a = 8
• b = 8
• C = ∠ABC = 120°

Substituting values into the formula:

`\implies \frac12\cdot 8 \cdot 8 \sin(120)=16√(3) \ \sf cm^2`

Answer 2

a) 120°

b) 16√3 ≈ 27.7 cm²

Explanation:

a)

Each exterior angle of a regular polygon measures 360° divided by the number of sides. For a hexagon, the exterior angle is 360°/6 = 60°. That means the adjacent interior angle is 180°-60° = 120°.

Angle ABC measures 120°.

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b)

The area of triangle ABC can be found using the formula ...

A = 1/2ab·sin(C) . . . . area of triangle with sides a, b, and angle C between them

A = 1/2(8 cm)(8 cm)sin(120°) = (32 cm²)(√3/2) = 16√3 cm²

The area of the triangle is 16√3 cm² ≈ 27.7 cm².