Answer: 204.12 inches
Step-by-step explanation:
We can find the lengths of the unknown sides by applying these identities:
[tex]tan\alpha=\frac{opposite}{adjacent}\\\\sin\alpha=\frac{opposite}{hypotenuse}[/tex]
Observe the image attached. To find "a" we need to substitute the following values into [tex]tan\alpha=\frac{opposite}{adjacent}[/tex]:
[tex]\alpha=30\°\\opposite=x=27\\adjacent=a[/tex]
And solve for "a":
[tex]tan(30\°)=\frac{27}{a}\\\\a=\frac{27}{tan(30\°)}\\\\a=46.76\ inches[/tex]
To find "b" we need to substitute the following values into [tex]sin\alpha=\frac{opposite}{hypotenuse}[/tex]:
[tex]\alpha=30\°\\opposite=x=27\\hypotenuse=b[/tex]
And solve for "b":
[tex]sin(30\°)=\frac{27}{b}\\\\b=\frac{27}{sin(30\°)}\\\\b=54 inches[/tex]
To find "c" we need to substitute the following values into [tex]sin\theta=\frac{opposite}{hypotenuse}[/tex]:
[tex]\theta=45\°\\opposite=x=27\\hypotenuse=c[/tex]
And solve for "c":
[tex]sin(45\°)=\frac{27}{c}\\\\c=\frac{27}{sin(45\°)}\\\\c=38.18 inches[/tex]
Since the triangle on the left is Isosceles, then:
[tex]d=c= 38.18\ inches\\\\[/tex]
Therefore, the perimeter is:
[tex]P=(46.76+54+2(38.18)+27)\ inches=204.12\ inches[/tex]
