Two of the sides of a triangular lot have lengths of 90 feet and 80 feet. If the angle opposite the 80-foot side is 59°, what is the length of the third side of the lot to the nearest tenth of a foot?

Looks like the ambiguous case of the sine law.

let the angle opposite the 90 be Ø
sinØ/90 = sin 59/80
sinØ =.9643
Ø = 74.6 or 105.4°

so the third angle is
46.4° or 15.6°

now use the sine law for each case to find the two different possibilities for the third side.

There are two triangles possible with your given information, try sketching both using my answers.

Jason designed an arch made of wrought iron for the top of a mall entrance. The 11 segments between the two concentric circles are each 1.25 m long. Find the total length of wrought iron used to make the structure. Round the answer to the nearest meter. Show your work.

To find the length of the third side of the triangular lot, we can use the Law of Cosines. The formula is as follows:

c^2 = a^2 + b^2 - 2ab * cos(C)

Where:
- c is the length of the third side
- a and b are the lengths of the other two sides
- C is the angle opposite side c

Given that a = 90 ft, b = 80 ft, and C = 59°, we can substitute these values into the formula:

c^2 = 90^2 + 80^2 - 2(90)(80) * cos(59°)

Simplifying:

c^2 = 8100 + 6400 - 2(90)(80) * cos(59°)

c^2 = 14500 - 14400 * cos(59°)

Now, let's calculate the value of cos(59°):

cos(59°) ≈ 0.5736

Substituting this value back into the equation:

c^2 = 14500 - 14400 * 0.5736

c^2 ≈ 14500 - 8270.4

c^2 ≈ 6230.6

Taking the square root of both sides, we get:

c ≈ √6230.6

c ≈ 78.9

Therefore, the length of the third side of the triangular lot is approximately 78.9 feet, rounded to the nearest tenth.

To find the length of the third side of the triangular lot, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c, and the opposite angles A, B, and C respectively, the following equation holds true:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, we know that side a has a length of 90 feet (opposite angle A), side b has a length of 80 feet (opposite angle B), and we are trying to find the length of side c (opposite angle C).

Using the Law of Cosines, we can find the length of side c:

c^2 = 90^2 + 80^2 - 2 * 90 * 80 * cos(59°)

c^2 = 8100 + 6400 - 14400 * cos(59°)

Now we can calculate the value of c^2:

c^2 = 8100 + 6400 - 14400 * cos(59°)

Using a calculator, evaluate cos(59°):

cos(59°) ≈ 0.5141

Now substitute this value into the equation:

c^2 = 8100 + 6400 - 14400 * 0.5141

Simplify the equation:

c^2 = 8100 + 6400 - 7394.4

c^2 = 1705.6

To find the length of side c, take the square root of 1705.6:

c = √1705.6

Using a calculator, find the square root:

c ≈ 41.3 feet

Therefore, the length of the third side of the lot is approximately 41.3 feet.