38.8k views
2 votes
CAN SOMEONE HELP ME PLEASSEEEEE

CAN SOMEONE HELP ME PLEASSEEEEE-example-1

1 Answer

1 vote

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)

User Nick Ludlam
by
8.1k points

Related questions

asked Oct 27, 2024 229k views
JimmyD asked Oct 27, 2024
by JimmyD
8.1k points
1 answer
2 votes
229k views
asked Jun 16, 2024 160k views
Workhardcc asked Jun 16, 2024
by Workhardcc
7.1k points
1 answer
0 votes
160k views