Rectangles Gepmetry

**Dragon** · January 10th 2007, 03:22 PM

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?

**Quick** · January 10th 2007, 03:54 PM

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 $b_1$

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

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: $3b_1h_1=b_2h_2\quad\Longrightarrow\quad3b_1h_1=135 0\quad\Longrightarrow\quad b_1h_1=450$ (make sure to check my arithmetic)

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

Also: $b_1=h_2$

Substitute: $b_1=2h_1$

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

Substitute: $450=2h_1^2$

Divide: $225=h_1^2$

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

Now you know that: $2h_1=h_2$

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

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

Substitute: $30b_2=1350$

Divide: $b_2=45$

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

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

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

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

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

**Soroban** · January 10th 2007, 05:01 PM

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 $x$ = length of the small rectangles
and $y$ = width of the small rectangles.

Code:

           y            x
      * - - - - + - - - - - - - *
      |         |               |
      |         |               | y
      |         |               |
    x |         + - - - - - - - +
      |         |               |
      |         |               | y
      |         |               |
      * - - - - + - - - - - - - *
           y            x

Since the left and right sides are equal: $x = 2y$
. . and the rectangle looks like this:

Code:

                   3y
      * - - - - - - - - - - - - *
      |                         |
      |                         |
      |                         |
   2y |                         | 2y
      |                         |
      |                         |
      |                         |
      * - - - - - - - - - - - - *
                   3y

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

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

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

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

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