Answer:
583 cm²
Step-by-step explanation:
To find the area of triangle BCD, we need to know the length of segment BD. That can be found from the law of sines:
BD/sin(A) = BA/sin(D)
BD = (sin(A)/sin(D))BA = sin(70°)/sin(74°)·(39 cm) ≈ 38.1249 cm
Then the area of BCD is ...
A = 1/2(DB)(DC)sin(BDC)
Angle BDC, together with the marked angles, makes a total of 180°.
∠BDC = 180° -74° -70° = 36°
A = 1/2)(38.1249 cm)(52 cm)sin(36°) ≈ 582.64 cm² ≈ 583 cm²
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Additional comment
The angle relation we used comes from the fact that consecutive interior angles where a transversal crosses parallel lines are supplementary. That means angle DAB and ADC are supplementary. Angle ADC is the sum of angles ADB and BDC, so we have ...
∠DAB +∠ADB +∠BDC = 180°
∠BDC = 180° -∠DAB -∠ADB
