Question

What is the area of ΔABC given m∠B = 115°, side A = 8 feet, and side C = 13 feet?

Answer 1

The area of ΔABC can be calculated using the formula for the area of a triangle:
`\(\text{Area} = (1)/(2) * \text{base} * \text{height}\).`

Step-by-step explanation:

Given that angle
`\(B = 115^\circ\),`side
`\(A = 8\)`feet, and side
`\(C = 13\) feet,` we can use the trigonometric relationship of the sine function to find the area of the triangle. First, we need to find the height of the triangle corresponding to side \(A\) using the sine function: \(\sin B = \frac{\text{opposite side}}{\text{hypotenuse}}\). Therefore,
`\(\sin 115^\circ = \frac{\text{height}}{13}\).` Rearranging this equation, we get the height of the triangle, which is
`\(\text{height} = 13 * \sin 115^\circ\).`

Now that we have the height and the base (side \(A\) = 8 feet), we can calculate the area of the triangle using the formula for the area of a triangle:
`\(\text{Area} = (1)/(2) * \text{base} * \text{height}\).`Substituting the values, we get
`\(\text{Area} = (1)/(2) * 8 * (13 * \sin 115^\circ)\).`

After calculating the height using trigonometric functions, the final formula for the area becomes
`\(\text{Area} = 4 * 13 * \sin 115^\circ\),` resulting in the area of ΔABC as \
`(52 * \sin 115^\circ\)`square feet.

This approach utilizes trigonometric principles to determine the height of the triangle and subsequently computes the area based on the given side lengths and angle measure, resulting in an accurate calculation of the triangle's area.

"By employing trigonometric relationships and the area formula for triangles, the calculation accurately derives the area of ΔABC based on the given angle measure and side lengths."

Answer 2

The area of triangle ABC is approximately 47.13 square feet.

Here's how to find the area:

1. Identify the formula: We can use the formula for the area of a triangle given two sides and the included angle:

Area = (1/2) * side1 * side2 * sin(angle)

2. Plug in the values: In this case, side1 (A) = 8 feet, side2 (C) = 13 feet, and the included angle (B) = 115°. Remember to convert the angle to radians before using the sine function (because most calculators use radians). So, 115° * π / 180 ≈ 2.0071 radians.

3. Calculate the area:

Area = (1/2) * 8 feet * 13 feet * sin(2.0071) ≈ 47.13 square feet