Question

Find the area of NPO

Answer 1

We have to calculate the side NO so that to apply Heron's formula.
To calculate NO, let's find the angles of Δ NOP:
∠ PNO = 180°-142° = 38°
∠NPO = 180°- 38° -84° ; ∠ NPO = 58°

1) To calculate NO, apply the cosine law: a²=b²+c²-2bc.cos(Ф)

 (NO)² = NP² + PO² - 2.NP.PO.cos 58°
  NO² = 26² + 16² -2.(26)(16).cos 58°
 NO² = 491.10 & NO = √491.1 & NO = 22.16

2) Now we have the 3 sides: NP=26, OP=16, & ON=22.16
 Perimeter = 26+16+22.16 = 64.16.
HALF of the perimeter = 64.16/2 =32.08 , this is p in Heron's formula:
AREA = √p(p-a)(p-b)(p-c)
AREA = √[32.08(32.08-26)(32.08-16)(32.08-22.16)]
AREA = √(31112.55) 
AREA = 176.38 ≈ 176.4 unit² ( answer C)