Question

In the figure, the ratio of the perimeter of rectangle ABDE to the perimeter of triangle BCD is . The area of polygon ABCDE is square units.

Choices for the first blank.
A: 2/3
B: 1
C: 3/2
D: 2

Choices for the second blank
A: 18
B: 22
C: 24
D: 28

Answer 1

Step 1

Find the perimeter of rectangle ABDE

we know that

the perimeter of rectangle is equal to

`P=2b+2h`

In this problem

`b=ED=2\ units`

`h=AE=6\ units`

substitute

`P=2*2+2*6=16\ units`

Step 2

Find the perimeter of triangle BCD

we know that

the perimeter of triangle is equal to

`P=BD+DC+BC`

In this problem we have

`BD=AE=6\ units`

`DC=BC`

Applying the Pythagoras theorem

`DC^(2)=4^(2)+3^(2)`

`DC^(2)=25`

`DC=5\ units`

substitute

`P=6+5+5=16\ units`

Find the ratio of the perimeter of rectangle ABDE to the perimeter of triangle BCD

we have

the perimeter of rectangle is equal to

`P=16\ units`

the perimeter of the triangle is

`P=16\ units`

so

the ratio is equal to

`(16)/(16) =1`

therefore

the answer Part 1) is the option B

`1`

Step 3

Find the area of polygon ABCDE

we know that

The area of polygon is equal to the sum of the area of rectangle plus the area of triangle

Area of rectangle is equal to

`A=AE*BD=6*2=12\ units^(2)`

Area of the triangle is equal to

`A=(1)/(2)AEh`

the height h of the triangle is equal to
`4\ units`

substitute

`A=(1)/(2)(6)(4)=12\ units^(2)`

The area of polygon is

`12\ units^(2)+12\ units^(2)=24\ units^(2)`

therefore

the answer part 2) is the option C

`24\ units^(2)`