Question

the area of triangle efg is 34.8 square units. what is the perimeter of triangle efg to the nearest unit?
29 units
30 units
41 units
60 units
trigonometric area formula: area = $\frac{1}{2}absin(c)$

Explanation:

Step1: Identify sides and angle for area formula

Let $a = 10$, $b = 12$, $\angle C=56^{\circ}$. The area formula is $A=\frac{1}{2}ab\sin(C)$.

Step2: Check if area calculation is consistent

First, calculate $\frac{1}{2}\times10\times12\times\sin(56^{\circ})$.
We know that $\sin(56^{\circ})\approx0.829$. Then $\frac{1}{2}\times10\times12\times0.829 = 60\times0.829 = 49.74$. But we are given area $A = 34.8$. There seems to be a mis - understanding in the problem setup. Let's assume we use the given area to find the included angle correctly.
$34.8=\frac{1}{2}\times10\times12\times\sin(C)$
$\sin(C)=\frac{34.8\times2}{10\times12}=\frac{69.6}{120}=0.58$
$C=\sin^{- 1}(0.58)\approx35.4^{\circ}$

Step3: Use the Law of Cosines to find the third side

The Law of Cosines is $c^{2}=a^{2}+b^{2}-2ab\cos(C)$.
Substitute $a = 10$, $b = 12$, and $C\approx35.4^{\circ}$. $\cos(35.4^{\circ})\approx0.814$.
$c^{2}=10^{2}+12^{2}-2\times10\times12\times0.814$
$c^{2}=100 + 144-195.36$
$c^{2}=48.64$
$c=\sqrt{48.64}\approx7$

Step4: Calculate the perimeter

The perimeter $P=a + b + c$.
$P=10 + 12+7=29$

Answer:

29 units