Geometry

A farmer is estimating the surface area of his barn to find how much paint he needs to buy. One part of the barn is triangular as shown.

The base of the triangle is 22 meters long
Both angles on either side connecting the base to each leg is 30 degrees. (It looks like an isosceles triangle)

a. The darkened sides in the figure are the edges of the roof. This trim will be painted white. Find the length of each of these two sides of the triangle. Explain how you found the answer.

b. The triangular surface will be painted red. Find the area of the triangle. Explain how you found the answer.

I know it has something to do with using cosine... but I'm not sure how to do it. Step by step would be the best!:) Please? Thank you!

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  1. By the sine law, we know that

    sin(<A)/A is sin(<B)/B which is equal to sin(<C)/C.

    THEREFORE,
    sin(<120)/22m = sin(<30)/x

    We need to isolate x.

    x= 22m * (sin 30 degrees/sin 120 degrees)
    x=12.7017059222m

    Since we have an isosceles triangle, each side is about 12.7m.

    PART B

    We need to find the area of the triangle.

    The formula for area is base x height /2 .

    Our height we need to find out with the pythagorean theorem. half of our triangle means a length of the triangle at 11m.

    a^2+b^2=c^2.

    Therefore, (12.7)^2=11^2+x^2

    Our height is 6.35084703891 m.

    Base x height /2= area
    area= 6.35084703891 m * 11 m / 2= 34.929658714m.

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