Answer:
A = 54√3 + 54 (exact)
A = 147.5 (approximate)
perimeter = 30 + 6√3 + 6√6 (exact)
perimeter = 55.1 (approximate)
Explanation:
We are looking for two things:
(a) area of a trapezoid
(b) perimeter of a trapezoid
(a)
The area of a trapezoid is given by the formula
A = (1/2)(B + b)h
where B and b are the lengths of the parallel bases.
h = height (perpendicular distance between bases)
Draw a segment from point C to segment AB and perpendicular to segment B. Call the point of intersection E.
Triangle BCE is a 30-60-90 triangle.
BE is the short leg. CE is the long leg. BC is the hypotenuse.
CE is the height of the trapezoid, h.
The ratio of the lengths of the sides of a 30-60-90 triangle is:
short leg : long leg : hypotenuse
1 : √3 : 2
In triangle BCE, the sides are:
BE = short leg
CE = long leg
BC = hypotenuse
BC:BE = 2:1
12/BE = 2
2BE = 12
BE = 6
CE:BE = √3:1
CE/BE = √3
CE = BE × √3
CE = 6√3
Now drop a perpendicular from point A to the bottom base, CD. Let the point of intersection on CD be called F.
Triangle ADF is a 45-45-90 triangle. That makes sides FA and DF congruent.
DF = FA = CE = 6√3
On the upper base, since BE = 6, and AB = 12, then AE = 6.
AECF is a rectangle, so FC = AE = 6.
CD = FC + DF
CD = 6 + DF
We need to find DF.
Triangle ADF is a 45-45-90 triangle.
FA and DF are the congruent legs. AD is the hypotenuse.
The ratio of the lengths of the sides of a 45-45-90 triangle is:
leg : leg : hypotenuse
1 : 1 : √2
DF = 6√3
AD:DF = √2:1
AD/(6√3) = √2
AD = 6√6
CD = CF + DF
CD = 6 + 6√3
upper base: AB = 12
lower base: CD = 6 + 6√3
height = CE = 6√6
A = (1/2)(B + b)h
A = (1/2)(CD + AB)(CE
A = (1/2)(6 + 6√3 + 12)(6√3)
A = (9 + 3√3)(6√3)
A = 54√3 + 54 (exact)
A = 147.5 (approximate)
(b)
The perimeter of a trapezoid is the sum of the lengths of the 4 sides.
AB = 12
BC = 12
CD = 6 + 6√3
We need the length of side AD.
AD:DF = √2:1
AD = DF × √2
AD = 6√3 × √2
AD = 6√6
perimeter = AB + BC + CD + AD
perimeter = 12 + 12 + 6 + 6√3 + 6√6
perimeter = 30 + 6√3 + 6√6 (exact)
perimeter = 55.1 (approximate)