The area of a triangle is $2632$ . Two of the side lengths are $85$ and $63$ and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of $a$ degree. $\newline$ Answer:

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Q. The area of a triangle is

2632

. Two of the side lengths are

85

and

63

and the included angle is obtuse. Find the measure of the included angle, to the nearest tenth of

a

degree.

\newline

Answer:

Area Formula: The area of a triangle can be calculated using the formula: $\newline$ Area = $(1/2) \times a \times b \times \sin(C)$ $\newline$ where $a$ and $b$ are the lengths of two sides, and $C$ is the included angle between them. We are given that the area is $2632$ , and the side lengths are $85$ and $63$ . We need to find the measure of the included obtuse angle $C$ .
Plug Known Values: First, let's plug the known values into the area formula: $\newline$ $2632 = \frac{1}{2} \times 85 \times 63 \times \sin(C)$ $\newline$ Now, we need to solve for $\sin(C)$ .
Isolate $\sin(C)$ : To isolate $\sin(C)$ , we multiply both sides of the equation by $2$ and then divide by the product of the side lengths $(85 \times 63)$ : $\sin(C) = \frac{2 \times 2632}{85 \times 63}$ $\sin(C) = \frac{5264}{5355}$
Calculate $\sin(C)$ : Now, we calculate the value of $\sin(C)$ : $\sin(C) = \frac{5264}{5355}$ $\sin(C) \approx 0.9828$ Since the angle is obtuse, we know that $\sin(C)$ will be positive and $C$ will be greater than $90$ degrees but less than $180$ degrees.
Find Angle C: To find the angle $C$ , we need to take the inverse sine (arcsin) of $\sin(C)$ . However, since the range of arcsin is typically from $-90$ to $90$ degrees, and we know that $C$ is obtuse, we will use the fact that $\sin(180° - C) = \sin(C)$ to find the correct angle in the obtuse range.
Calculate Angle C: We calculate the angle C using a calculator set to degree mode: $\newline$ $C \approx 180° - \arcsin(0.9828)$ $\newline$ $C \approx 180° - 79.1°$ $\newline$ $C \approx 100.9°$