Congruent Perimeter

Geometry Level 1

A rectangle of perimeter 44 is partitioned into 5 congruent rectangles as indicated in the diagram. The perimeter of each of the congruent rectangle is

Details and Assumptions :

Figure not to scale.

20 10 48 24

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Tanveen Dhingra by tanveen dhingra , 19, India

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6 solutions

Gamal Sultan
Feb 11, 2015

Let the length of one of the small rectangles be x

Let the width of one of the small rectangles be y

Then

The length of the big rectangle = 3 y

The width of the big rectangle = x + y

Then

8 y + 2 x = 44 ...............................(1)

But, from the diagram

2 x = 3 y .........................................(2)

From (1), (2)

11 y = 44

y = 4

From (2)

x = 6

Then

The perimeter of each small rectangle = 2(6 + 4) = 20

26 Helpful 0 Interesting 0 Brilliant 0 Confused

length=x u said then why 3y instead of 3x?

Pushan Paul - 6 years, 4 months ago

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Length here means the longer side of the rectangle, not the side of the rectangle that is parallel to the x-axis in a coordinate plane. Therefore, width here means the shorter side of the rectangle.

x is the length of one of the small rectangles, while y is the width of one of the small rectangles. If you pay close attention to the left side of the big rectangle, it is equal to 3y, or 3 of the width of the small rectangles.

Bisma Joyosumarto - 6 years, 3 months ago

I like it.

Mashiur Rahman - 5 years, 1 month ago
Rich Moraghan
Feb 15, 2015

2 long sides (A) = 3 short sides (B) ---> 2A = 3B ---> A = (3/2)B

Perimeter Total (Pt) = 4A + 5B = 44

Perimeter Small (Ps) = 2A + 2B

Pt = 4(3/2)B + 5B = 44 ---> 11B = 44 ---> B = 4

A = (3/2)4 ---> A =6

Ps = 2A + 2B ---> 2(6) + 2(4) = 20

3 Helpful 0 Interesting 0 Brilliant 0 Confused

There are 5 short sides (x) and 4 long sides (y)which make up the perimeter of 44 units.

5x + 4y = 44

A single congruent rectangle's perimeter is 2 long sides and 2 short.

2x + 2y = ???

2x is approx. 0.5 (5x) and 2y is 0.5 (4y). If everything halved exactly (ie if 4x + 4y = 44), the answer would be 22. 4x + 4y = 44 (÷2) --> 2x + 2y = 22

However, using the above equations:

5x + 4y = 44 (2) --> 2.5x + 2y = 22 2x + 2y < 2.5x + 2y Therefore the answer will be less than 22, but not by much

Therefore perimeter of congruent rectangle = 20 (closest answer which is less than 22).

(This is just educated guesswork, I've not done formal mathematics since AS Levels 😂)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You can simplify to 6long and 2short 44=6L+2S Then the short side is 1/3 size of a long or L=1.5S Substitute so all in short 44=(6x1.5=9)+2S=11S so S=4 L=1.5xS=6 Check your maths (6x6)+(2x4)=44 Then small rectangle = 2L+2S (2x6)+(2x4)=20

Laura Quint - 4 years, 4 months ago

Very nice sir

Lal Bahadur - 3 years, 6 months ago
Dae De Vibar
Feb 16, 2015

Ahhhmmm

let x=width, y=length

2x+2y = 44

so let's put width=10 length=12 (any may do tho);

2x+2y=44

2(10) + 2(12) = 44

20 + 24 = 44

ok?

so with that, let's get the area.

xy=A

(10)(12)= A

A= 120

so from 120 we divide it by 5 since congruence is absolute.

120/5 = 24

and from 24 (the area of the smaller rectangles), we get its factors which are all possible dimensions of the congruent rectangle.

XY (1)(24), perimeter 50; (2)(12), perimeter 28; (3)(8), perimeter 22; (4)(6), perimeter 20;

so why 20 is the answer? because I felt it. Joke, they are factors closest to 5, the point of congruence.

(I just came here to share. I don't really know what I'm talking about but heyy I got it correct!)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Istiak Reza
Feb 11, 2015

Let the length of the bigger rectangle be x and width be y. we get 2( x + y)= 44 ⇒ x + y = 22 let the length of the smaller rectangles be a and width be b so, x=3a and y=a+b If we place the 5 rectangles in a row in Such a way that the width of each rectangle touches the other, we find the area of the new rectangle to be ×Y = 5 ab therefore,x= 5ab/y = 5ab/a+ b putting ×=3a,we find b=( 3/2) a putting the values of X and y in the equation ×+y=22 and putting ( 3/2) a for b,we get a=4 and b = (3/ 2) X 4 = 6 therefore the perimeter of each smaller rectangle is 2( 4+6) = 20

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Since what is given in the sketch does not give any of the given answer, let us try stacking the small ones on one another. Let small ones be X*Y. So the perimeter of the big is 2(X+5Y) = 44. Y=3 and so X=7. Smaller perimeter is 2(7 + 3)=20.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Interesting, but my concern is that the equation 2 ( X + 5 Y ) = 44 2(X + 5Y) = 44 does not have a unique solution. One solution is Y = 3 , X = 7 Y = 3, X = 7 , giving a perimeter for the small rectangles of 20 20 . But other possible integral solutions include Y = 4 , X = 2 Y = 4, X = 2 , giving a perimeter for the small rectangles of 12 12 . and Y = 2 , X = 12 Y = 2, X = 12 , giving a perimeter of 28 28 . So I think that the original configuration is a critical piece of information in the process of determining of a unique solution.

Brian Charlesworth - 6 years, 4 months ago

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Greeting. I agree with you. In fact I too would have come up with the same solution as your, had I not made calculation mistake, which made me see the solution I gave!!

Niranjan Khanderia - 6 years, 4 months ago

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