What is the area of this triangle?

enter your answer as a decimal in the box. round only your final answer to the nearest tenth.

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

The Principle of Zero Products states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0. Once the polynomial is factored, set each factor equal to zero and solve them separately.