In ΔQRS, r = 510 cm, ∠S=5° and ∠Q=15°. Find the area of ΔQRS, to the nearest centimeter.

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.