Answer:
13.2
Step-by-step explanation:
First we find the missing side of the triangle using the law of cosines:
c² = a²+b²-2ab(cos C)
c² = 8²+7²-2(8)(7)(cos 28)
c² = 64+49-112(cos 28)
c² = 113-112(cos 28)
c² = 14.1099
√c² = √14.1099
c = 3.8
Next we use the law of sines to find the angle opposite the 7 cm side:
sin 28/3.8 = sin B/7
Cross multiply:
7(sin 28) = 3.8(sin B)
Divide both sides by 3.8:
7(sin 28)/3.8 = 3.8(sin B)/3.8
0.8648 = sin B
Take the inverse sine:
sin⁻¹(0.8648) = sin⁻¹(sin B)
60 = B
The height of the triangle will extend from the lowest vertex to the 8 cm side. This makes the right triangle formed to the left a 30-60-90 triangle. In these triangles, the side opposite the 90 degree angle is twice as much as the side opposite the 30 degree angle; this means the portion of the 8 cm side from the height to the left will be 3.8/2 = 1.9 cm.
Using the Pythagorean theorem with 1.9 as a leg and 3.8 as the hypotenuse,
1.9² + b² = 3.8²
3.61 + b² = 14.44
Subtract 3.61 from each side:
3.61 + b² - 3.61 = 14.44 - 3.61
b² = 10.83
Take the square root of each side:
√b² = √10.83
b = 3.3
This means the height of the triangle is 3.3. Using 8 as the base,
A = 1/2bh
A = 1/2(3.3)(8) = 13.2
