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!!!!!!!!!!!
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: .$\displaystyle 37.5^2\:=\:\boxed{1406.25\text{ cm}^2}$