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Involves sine/cosine rules

Involves sine/cosine rules-example-1
User CSharpRocks
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1 Answer

22 votes
22 votes

Answer:

44.47 cm² (nearest hundredth)

Explanation:

Area of ΔABC = 1/2 x base x height

⇒ 21 = 1/2 x 7 x BC

⇒ BC = 6 cm

Pythagoras' Theorem: a² + b² = c²

(where a and b are the legs, and c is the hypotenuse, of a right triangle)

⇒ AB² + BC² = AC²

⇒ 7² + 6² = AC²

⇒ AC² = 85

⇒ AC = √85 cm

Cosine rule to find length AD:

c² = a² + b² - 2 ab cosC

⇒ DC² = AD² + AC² - 2(AD)(AC)cos(DAC)

⇒ 9.2² = AD² + (√85)² - 2(AD)(√85)cos 73°

⇒ AD² - 5.39106...AD + 0.36 = 0

⇒ AD = 5.323442445, 0.06762541414

⇒ AD = 5.323442445

Area of a triangle ADC: (1/2)absinC

(where a and b are adjacent sides and C is the angle between them)

⇒ area = (1/2) × AC × AD × sin(DAC)

⇒ area = (1/2) × √85 × 5.323442445 × sin(73°)

⇒ area =23.4675821... cm²

Area of quadrilateral = area of ΔABC + area of ΔADC

= 21 + 23.4675821...

= 44.47 cm² (nearest hundredth)

User Thiago Falcao
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