High School

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------------------------------------------------ Find the area of the triangle given:

- \( C = 133.6^\circ \)
- \( a = 46.1 \, \text{ft} \)
- \( b = 36.6 \, \text{ft} \)

Answer :

The area of the triangle is approximately 531.9 square feet.

  • Identify the given values and the formula for the area of a triangle using two sides and the included angle:

[tex]\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \][/tex]

Given: [tex]\( a = 46.1 \)[/tex] ft, [tex]\( b = 36.6 \)[/tex] ft, [tex]\( C = 133.6^\circ \).[/tex]

  • Convert the angle to radians if needed:

[tex]\[ \text{Angle in radians} = 133.6^\circ \times \frac{\pi}{180} \approx 2.331 \, \text{radians} \][/tex]

  • Calculate the area using the given values:

[tex]\[ \text{Area} = \frac{1}{2} \times 46.1 \times 36.6 \times \sin(133.6^\circ) \][/tex]

[tex]\[ \sin(133.6^\circ) \approx 0.6691 \][/tex]

[tex]\[ \text{Area} \approx \frac{1}{2} \times 46.1 \times 36.6 \times 0.6691 \approx 531.9 \, \text{square feet} \][/tex]

To find the area of the triangle with given measurements C=133.6°, a=46.1 ft, and b=36.6 ft, use the formula Area = (1/2) * a * b * sin(C). Converting the angle to radians and calculating using a calculator gives the area as approximately 612.49 square feet.

To find the area of a triangle when two sides and the included angle are known, we use the formula:

Area = (1/2) * a * b * sin(C)

Given the values:

  • a = 46.1 ft
  • b = 36.6 ft
  • C = 133.6°

First, convert the angle C from degrees to radians, as trigonometric functions typically use radians:

C (in radians) = 133.6° * (π/180)

C ≈ 2.331 radians

Now, plug the values into the formula:

Area = (1/2) * 46.1 * 36.6 * sin(2.331)

Using a calculator to find sin(2.331):

sin(2.331) ≈ 0.728

So the area is:

Area ≈ (1/2) * 46.1 * 36.6 * 0.728

Area ≈ 612.49 square feet

Thus, the area of the triangle is approximately 612.49 square feet.