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1. Find the area of triangle ABC. Given: a = 75 cm, b = 40 cm, C = 120°. Round to three significant digits.
2. Find the area of triangle ABC. Given: a = 40.9 cm, c = 49.1 cm, B = 153.9°. Round to three significant digits.
3. Find the area of triangle ABC. Given: b = 0.919 cm, c = 0.671 cm, A = 67°50'. Round to three significant digits.
4. Find the area of a parallelogram with one angle of 60° and sides of 15 inches and 13 inches. Round to the nearest whole number.

Answer :

To find the area of a triangle, when given two sides and the included angle, you can use the formula:

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

Let's solve each part of the question step-by-step:

1. Find the area of triangle ABC.

  • Given: [tex]a = 75[/tex] cm, [tex]b = 40[/tex] cm, and [tex]C = 120^\circ[/tex].
  • Use the formula:

[tex]\text{Area} = \frac{1}{2} \times 75 \times 40 \times \sin(120^\circ)[/tex]

  • [tex]\sin(120^\circ)[/tex] is the same as [tex]\sin(60^\circ) = \frac{\sqrt{3}}{2}[/tex].

[tex]\text{Area} = \frac{1}{2} \times 75 \times 40 \times \frac{\sqrt{3}}{2}[/tex]

[tex]\text{Area} = 975\sqrt{3} \approx 1687.52 \text{ cm}^2[/tex]

  • Rounded to three significant digits: 1680 cm²

2. Find the area of triangle ABC.

  • Given: [tex]a = 40.9[/tex] cm, [tex]c = 49.1[/tex] cm, and [tex]B = 153.9^\circ[/tex].
  • Use the formula:

[tex]\text{Area} = \frac{1}{2} \times 40.9 \times 49.1 \times \sin(153.9^\circ)[/tex]

  • Calculate [tex]\sin(153.9^\circ)[/tex] using a calculator.

[tex]\text{Area} = \frac{1}{2} \times 40.9 \times 49.1 \times 0.4067 \approx 407.5461 \text{ cm}^2[/tex]

  • Rounded to three significant digits: 408 cm²

3. Find the area of triangle ABC.

  • Given: [tex]b = 0.919[/tex] cm, [tex]c = 0.671[/tex] cm, and [tex]A = 67^\circ50'[/tex].
  • First, convert [tex]67^\circ50'[/tex] to decimal degrees.

[tex]67^\circ50' = 67 + \frac{50}{60} = 67.8333^\circ[/tex]

  • Use the formula:

[tex]\text{Area} = \frac{1}{2} \times 0.919 \times 0.671 \times \sin(67.8333^\circ)[/tex]

  • Calculate [tex]\sin(67.8333^\circ)[/tex] using a calculator.

[tex]\text{Area} \approx \frac{1}{2} \times 0.919 \times 0.671 \times 0.9272 \approx 0.28677 \text{ cm}^2[/tex]

  • Rounded to three significant digits: 0.287 cm²

4. Find the area of a parallelogram.

  • Given: angle [tex]60^\circ[/tex] and sides 15 inches and 13 inches.
  • Use the formula for a parallelogram:

[tex]\text{Area} = \text{base} \times \text{height}[/tex]

  • Here, the formula modifies to:

[tex]\text{Area} = 15 \times 13 \times \sin(60^\circ)[/tex]

  • [tex]\sin(60^\circ) = \frac{\sqrt{3}}{2}[/tex]

[tex]\text{Area} = 15 \times 13 \times \frac{\sqrt{3}}{2} \approx 168.9[/tex]

  • Rounded to the nearest whole number: 169

In each case, the formulas for area involve the sine of the angle, which is crucial when working with non-right triangles or other polygons.

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