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Thread: Rectangles Gepmetry

  1. #1
    Junior Member Dragon's Avatar
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    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?
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  2. #2
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by Dragon View Post
    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!!!!!!!!!!!
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  3. #3
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    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: .$\displaystyle 37.5^2\:=\:\boxed{1406.25\text{ cm}^2}$

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