1. ## Rectangles Gepmetry

Three congruent rectangles are placed to form a larger rectangle as shown with an area of 1350cm^2 Find the area of a square that has the same perimeter as the larger rectangle?

2. Originally Posted by Dragon
Three congruent rectangles are placed to form a larger rectangle as shown with an area of 1350cm^2 Find the area of a square that has the same perimeter as the larger rectangle?
let's say the larger sides of the small rectangles are the bases of the small rectangles, we'll call the bases $\displaystyle b_1$

let's say the larger sides of the large rectangle are the bases of the large rectangle, we'll call those bases $\displaystyle b_2$

Same thing for height.

Now you know that the area of any rectangle is base times height, so since the three rectangles are congruent, you know that: $\displaystyle 3b_1h_1=b_2h_2\quad\Longrightarrow\quad3b_1h_1=135 0\quad\Longrightarrow\quad b_1h_1=450$ (make sure to check my arithmetic)

Look at the picture, you also know that: $\displaystyle 2h_1=h_2$

Also: $\displaystyle b_1=h_2$

Substitute: $\displaystyle b_1=2h_1$

Multiply both sides by $\displaystyle h_1$ to get: $\displaystyle b_1h_1=2h_1^2$

Substitute: $\displaystyle 450=2h_1^2$

Divide: $\displaystyle 225=h_1^2$

Find the square root of both sides: $\displaystyle h_1=15$

Now you know that: $\displaystyle 2h_1=h_2$

So: $\displaystyle 2(15)=h_2=30$

Alright, you also know that: $\displaystyle b_2h_2=1350$

Substitute: $\displaystyle 30b_2=1350$

Divide: $\displaystyle b_2=45$

So the perimeter of the Larger rectangle is: $\displaystyle 2b_2+2h_2=2(45)+2(30)=90+60=150$

So let's call the sides of the square $\displaystyle s$

Therefore: $\displaystyle 4s=150\quad\Longrightarrow\quad s=37.5$

So the area of the square is: $\displaystyle \boxed{37.5^2=1406.25}$

But MAKE SURE TO CHECK MY ARITHMETIC!!!!!!!!!!!

3. Hello, Dragon!

My solution is the same as Quick's . . . without the subscripts.

Three congruent rectangles are placed to form a larger rectangle
. . as shown with an area of 1350 cm^2
Find the area of a square that has the same perimeter as the larger rectangle.

Let $\displaystyle x$ = length of the small rectangles
and $\displaystyle y$ = width of the small rectangles.
Code:
           y            x
* - - - - + - - - - - - - *
|         |               |
|         |               | y
|         |               |
x |         + - - - - - - - +
|         |               |
|         |               | y
|         |               |
* - - - - + - - - - - - - *
y            x

Since the left and right sides are equal: $\displaystyle x = 2y$
. . and the rectangle looks like this:
Code:
                   3y
* - - - - - - - - - - - - *
|                         |
|                         |
|                         |
2y |                         | 2y
|                         |
|                         |
|                         |
* - - - - - - - - - - - - *
3y

Its area is: .$\displaystyle A \:=\:(3y)(2y) \:=\:6y^2$
Its perimeter is: .$\displaystyle P \:=\:3y + 2y + 3y + 2y \:=\:10y$

Since the area is 1350 cm², we have: .$\displaystyle 6y^2 \:=\:1350\quad\Rightarrow\quad y = 15$
And its perimeter is: .$\displaystyle P \:=\:10(15) \:=\:150$ cm.

The square with the same perimeter has a side of: $\displaystyle \frac{150}{4} = 37.5$ cm.

The area of this square is: .$\displaystyle 37.5^2\:=\:\boxed{1406.25\text{ cm}^2}$