Question

((Image Included))

If x = 27 inches, what is the perimeter of the figure above?
A. (108 + 54 + 54) inches
B. (54 + 108) inches
C. (364.5 + 729) inches
D. (27 + 54 + 81) inches

Answer 1

Answer: 204.12 inches

Explanation:

We can find the lengths of the unknown sides by applying these identities:

`tan\alpha=(opposite)/(adjacent)\\\\sin\alpha=(opposite)/(hypotenuse)`

Observe the image attached. To find "a" we need to substitute the following values into
`tan\alpha=(opposite)/(adjacent)`:

`\alpha=30\°\\opposite=x=27\\adjacent=a`

And solve for "a":

`tan(30\°)=(27)/(a)\\\\a=(27)/(tan(30\°))\\\\a=46.76\ inches`

To find "b" we need to substitute the following values into
`sin\alpha=(opposite)/(hypotenuse)`:

`\alpha=30\°\\opposite=x=27\\hypotenuse=b`

And solve for "b":

`sin(30\°)=(27)/(b)\\\\b=(27)/(sin(30\°))\\\\b=54 inches`

To find "c" we need to substitute the following values into
`sin\theta=(opposite)/(hypotenuse)`:

`\theta=45\°\\opposite=x=27\\hypotenuse=c`

And solve for "c":

`sin(45\°)=(27)/(c)\\\\c=(27)/(sin(45\°))\\\\c=38.18 inches`

Since the triangle on the left is Isosceles, then:

`d=c= 38.18\ inches\\\\`

Therefore, the perimeter is:

`P=(46.76+54+2(38.18)+27)\ inches=204.12\ inches`

Answer 2

=204.13 inches.

Explanation:

Using the side x, we can use sine to find the hypotenuse of the triangle with the angle marked 30°.

Sin 30 =x/hypotenuse.

sin 30 = 27/hyp

hyp= 27/sin 30

=54 inches

We can also find the adjacent as follows.

Cos 30 = adjacent/ 54

Adjacent= 54 cos 30

=46.77 inches

Using the angles marked 45 we can find the hypotenuse of the isosceles triangle.

sin 45= x/hypotenuse

sin 45 =27/hypotenuse

hypotenuse = 27/sin 45

=38.18 inches

The hypotenuse of both the triangles making the isosceles triangle are 38.18 inches long.

Perimeter = 54+ 46.77+27+ 38.18+38.18

=204.13 inches.