Answer: The area is 8,652 square centimeters (rounded off to the nearest centimeter)
Step-by-step explanation: The triangle has one side given and two angles given as well. The third angle can be calculated as follows;
Angle R = 180 - (5 + 15)
Angle R = 180 - 20
Angle R = 160
To calculate side s we shall apply the Sine rule.
s/SinS = r/SinR
s/Sin5 = 510/Sin160
By cross multiplication we now have
s = (510 x Sin5)/Sin160
s = (510 x 0.0872)/0.3420
s = 44.472/0.3420
s = 130
Next we calculate side q as follows;
q/SinQ = r/SinR
q/Sin15 = 510/Sin160
By cross multiplication we now have
q = (510 x Sin15)/Sin160
q = (510 x 0.2588)/0.3420
q = 131.988/0.3420
q = 385.929
q = 386 (approximately)
Having found all three sides of the triangle as 130, 386 and 510 respectively the area shall be calculated by use of the Heron’s formula which is;
A = [square root] s(s - a)(s - b)(s - c)
Where s is the semi-perimeter, a, b and c are the three sides. The semi-perimeter is calculated as follows;
s = (130 + 386 + 510)/2
s = 1026/2
s = 513
The area now becomes
A = [square root] 513(513 - 510)(513 - 386)(513 - 130)
A = [square root] 513(3)(127)(383)
A = [square root] 74858499
A = 8652.0806
Rounded to the nearest centimeter, the area of the triangle is 8,652 square centimeters.