Answer:
Area = 84.1 yd^2
Explanation:
Important Note:
For these kinds of problems, it's important to draw the triangle when solving. I'll attach a picture of a triangle I used to solve for the Area.
In my picture, the angle names S, T, and R are blue and the info we're already given (i.e., r = 10 yd, m∠S = 55°, m∠R = 29°
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Classifying △ STR:
- We know that this is a non-right triangle (it's obtuse) since it contains an angle greater than 90°.
Thus, we'll have to find the area using a non-right triangle sine area formula.
Sine formulas for the area of a non-right triangle:
- There are three non-right triangle sine area formulas, which allows us to solve for one value in the formula when we have the other three values.
Furthermore, the formulas use a, b, and c for sides and A, B, and C for angles:
Area = 1/2bc * sin (A)
Area = 1/2ac * sin (B)
Area = 1/2ab * sin (C)
Substituting s, t, r, S, T, and R for a, b, c, A, B, and C in the sine area formula:
When we rewrite the sine area formulas by substituting s, t, r, S, T, and R instead of a, b, c, A, B, and C, we have:
Area = 1/2tr * sin (S)
Area = 1/2sr * sin (T)
Area = 1/2st * sin (R)
Layout of the formula:
- In all the formulas, an angle is sandwiched between two sides.
Thus, we want either angle S to be sandwiched between r and t or angle T to be sandwiched between r and s.
- We can sandwich angle T between r and s by finding s.
Finding the measure of angle R:
- Before we can find s, we'll first need to find m∠R.
- The Triangle Sum Theorem states that the sum of a triangle's interior angles equals 180°.
Thus, we can find m∠R by subtracting the sum of m∠S (55°) and m∠T (96°) from 180:
m∠S + m∠T + m∠R = 180
m∠R = 180 - (m∠S + m∠T)
m∠R = 180 - (55 + 96)
m∠R = 180 - 151
m∠R = 29
Thus, m∠R = 29°.
Finding S using the Law of Sines:
To find S, we'll need to use the Law of Sines, which also has three forms:
a / sin (A) = b / sin (B) = c / sin (C)
As the proportion shows, we're able to find either one side or one angle if we have one completed ratio and either one angle in the proportion in which we're trying to find a side or one side in the proportion in which we're trying to find an angle.
Substituting s, t, r, S, T, and R for a, b, c, A, B, and C in the Law of Sines:
When we rewrite the law of sines by substituting s, t, and r, S, T, and R for a, b, c, A, B, and C in the Law of Sines, we have:
s / sin (S) = t / sin (T) = r / sin (R)
Finding s using the Law of Sines:
Since we now know that m∠R = 29°, we can use s / sin (S) = r / sin (R) and plug in 55 for S, 10 for r, and 29 for R to find s:
Step 1: Multiply both sides by sin (55) to find s:
(s / sin (55) = 10 / sin (29)) * sin (55)
s = 16.8963653
Thus, s = 16.8963653 yd.
It's best not to round until we get to the end so we find an exact value for the area. Then we'll round it to the nearest tenth.
Finding the Area using Area = 1/2sr * sin (T):
Now, we can plug in 16.8963653 for s, 10 for r, and 96 for T to find the area of △ STR. Then, we can round to the nearest tenth at the end to find the final answer:
Area △ STR = 1/2sr * sin (T)
Area △ STR = 1/2(16.8963653)(10) * sin (96)
Area △ STR = 84.48182648 * sin (96)
Area △ STR = 84.0190262
Area △ STR = 84.1
Thus, the area of △ STR is about 84.1 yd.