Question 1134281: A rectangle has a perimeter of 34cm. One of its diagonals is 13cm. Find its sides.
Found 3 solutions by rothauserc, Theo, greenestamps:
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! Let l be length and w be width, we know that
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2l +2w = 34
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1) l +w = 17
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By Pythagorean Theorem, we know that
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2) l^2 +w^2 = 13^2 = 169
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Solve equation 1 for l and substitute for l in equation 2
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l = 17-w
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(17-w)^2 +w^2 = 169
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289 -34w +w^2 +w^2 = 169
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2w^2 -34w +120 = 0
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divide both sides of = by 2
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w^2 -17w +60 = 0
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(w-12) * (w-5) = 0
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w = 12, w = 5
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We have two solutions for the rectangle
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1) length = 12 cm and width = 5 cm
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2) length = 5 cm and width = 12 cm
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Note usually the length is greater than the width for a rectangle
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Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the perimeter of the rectangle is 34 cm.
the formula for perimeter is 2x + 2y = p
when p = 34, the formula becomes 2x + 2y = 34
x is the length
y is the width
one of the diagonals is 13 cm.
since it's a rectangle, the other diagonal has to be 13 as well.
that's a nice fact to know but it's not necessary to know it to solve this problem.
the diagonal forms a right triangle with one of the sides being x and the other side being y.
by pythagorus, the diagonal squared is equal to x^2 + y^2.
the formula is therefore x^2 + y^2 = 13^2 which becomes x^2 + y^2 = 169.
you have two equations that needs to be solved simultaneously.
they are 2x + 2y = 34 and x^2 + y^2 = 169.
solve for y in the first equation to get y = 17 - x.
replace y with 17 - x in the second equation to get x^2 + (17 - x)^2 = 169
simplify to get x^2 + x^2 - 34x + 289 = 169
subtract 169 from both sides of that equation to get x^2 + x^2 - 34x + 120 = 0
combine like terms to get 2x^2 - 34x + 120 = 0
divide both sides of the equation by 2 to get x^2 - 17x + 60 = 0
factor that equation to get (x - 5) * (x - 12) = 0
that makes x = 5 or x = 12.
since y = 17 - x, .....
when x = 5, y = 12
when x = 12, y = 5
since x is the length of the rectangle, let x = 12.
you have length of the rectangle is 12 and width of the rectangle is 5.
the perimeter is equal to 2 * 12 + 2 * 5 = 24 + 10 = 35 cm which is correct.
x^2 + y^2 = 13^2 becomes 12^2 + 5^2 = 169 which becomes 144 + 25 = 169 which becomes 169 = 169 which is true.
the solution looks good.
the solution is that the length of the rectangle is 12 and the width of the rectangle is 5.
here's a good reference on the properties of a rectangle.
https://brilliant.org/wiki/properties-of-rectangles/
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Note that a formal algebraic solution is good exercise; but if the speed of solution is important and formal algebra is not required (as, for example, in a competitive math examination), this problem can be solved very quickly.
The length and width of the rectangle and the diagonal form a right triangle; and lengths of both the diagonal and the perimeter are integers. That means the three lengths of the sides of the triangle form a Pythagorean Triple.
A Pythagorean Triple with hypotenuse 13 is 5-12-13; if the side lengths are 5 and 12, then the perimeter is 2(5+12) = 34, as required.
So the side lengths are 5 and 12.
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