For the given we will find its and its one at a time. Let's do it!
Perimeter
The perimeter of a polygon is found by adding all of its side lengths.
We know the lengths of two sides of our figure, but we are missing the other two. Since the polygon has exactly one pair of sides, it is a . Also, since the nonparallel sides are , the trapezoid is an .
Let's draw another height to construct a second on the right-hand side of the diagram.
The right triangles have congruent and one pair of congruent legs. Therefore, by the the triangles are . Let x be the length of their base.
We can find the value of x by using the .
x+ 8+ x=12
Let's solve this equation.
The length of the base of each right triangle is 2 inches. Let's now pay close attention to just one of them.
We have a right triangle with an that measures 30^(∘) and whose opposite side length is 2 inches. To find the length of the hypotenuse — which is also a side of the trapezoid — we will use the .
sin θ = Opposite/Hypotenuse
If we let h be the length of the hypotenuse of the triangle, we can substitute the corresponding values in the above equation to find its length.
sin θ = Opposite/Hypotenuse
sin 30^(∘) = 2/h
sin 30^(∘) * h= 2
h= 2/sin 30^(∘)
h=4
The hypotenuse of the right triangle has a length of 4 inches. Note that the opposite side is congruent. Therefore, its length is also 4 inches.
We have all of the information we need to calculate the perimeter of the polygon.
Perimeter: 8+4+12+4=28in.
Finally, to convert 28 inches into feet, we will use the fact that 12 inches is equal to 1 foot. We have to multiply our answer by the 112.
Perimeter: 28* 1/12≈ 2.3ft
Area
The is one half the product of its base and the sum of its bases. In the given diagram, we can see that the lengths of the bases are 12 and 8 inches.
Let's use a right triangle again to find the height h.
The length of the hypotenuse is 4 inches and the length of one of the legs 2 inches. The length of the missing leg is also the height of the trapezoid. We can find it by substituting a= 2, b=h, and c= 4 into the . Let's do it!
a^2+b^2=c^2
2^2+h^2= 4^2
4+h^2=16
h^2=12
h=sqrt(12)
h=sqrt(4* 3)
h=sqrt(4)sqrt(3)
h=2sqrt(3)
Recall that h represents a side of a triangle and the height of the trapezoid. Therefore, we only kept the when solving the equation because h must be positive. The height of the trapezoid is 2sqrt(3) inches.
Having the height and the lengths of the bases, we can substitute them into the formula for the area of a trapezoid.
A=1/2h(b_1+b_2)
A=1/2(2sqrt(3))( 8+ 12)
A=1/2(2sqrt(3))(20)
A=1/2(2)(20)sqrt(3)
A=20sqrt(3)
The area of the trapezoid is 20sqrt(3)in.^2
To express it in square feet, we have to multiply it by 112 twice.
Area: 20sqrt(3)in.^2 * 1ft/12in. * 1ft/12in.
Finally, let's simplify this expression.
A=20sqrt(3)in.^2 * 1ft/12in. * 1ft/12in.
A=20sqrt(3)in.^2 * 1ft^2/144in.^2
A=20sqrt(3)in.^2 * 1ft^2/144in.^2
A=20sqrt(3) * 1ft^2/144
A=20sqrt(3) ft^2/144
A=20sqrt(3)/144ft^2
A=0.240562... ft^2
A≈ 0.2 ft^2
