the perimeter of a rectangle is 42 metres and 165 diagonal is 15 metres. what are the length of the sides?
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The perimeter of a rectangle is 2L+2W=P . The diagonals can be represented as L2+W2=D2 . Plugging in, we have:
2L+2W=42⟹L+W=21
L2+W2=152=225
We can square the first equation and subtract the second:
L+W=21
(L+W)2=212
L2+2LW+W2=441
L2+2LW+W2−(L2+W2)=441−225
2LW=216
It doesn’t tell us much, but we can now more easily substitute and solve for L .
W=21−L & LW=108
L(21−L)=108
21L−L2=108
L2−21L+108=0
We can use the quadratic formula
x=−b±b2−4ac√2a
L=21±441−432√2=21±9√2=21±32
L=21+32=242=12
We now know L=9 or L=12 . From this, we can calculate W by plugging L into W=21−L .
W=21−9=12
W=21−12=9
We have two solutions:
1)L=9 & W=12
2)L=12 & W=9
Both of which are the same, though. Our rectangle has side lengths of 9 , 12 , 9 , & 12 .
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