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The perimeter of a rectangle is 34 cm. Given that the diagonal is of length 13 cm, and that the width is x cm, derive the equation x¨2 - 17x + 60 =0. Hence find the dimensions.
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The perimeter is 34cm, and as the perimeter is made of 2 lengths and 2 widths, that means that length+width = 17.
We're also told that the diagonal is 13cm, and so by Pythagoras, length² + width² = 13² = 169.
We now have 2 simultaneous equations, so we need to eliminate one of the variables.
Rearranging the first equation shows that length = 17-width.
Substituting this into the second gives width² + (17-width)² = 169.
Expanding and replacing width by x turns this into x² + 289 - 34x + x² = 169.
Simplify: 2x² - 34x + 120 = 0
--> x² - 17x + 60 = 0.
From here, it's just a case of solving the quadratic equation that we now have.
x² - 17x + 60 = 0
(x-5)(x-12) = 0
x = 5 or 12.
By convention, length is generally larger than width, so the length is 12cm and the width is 5cm.
Why did the vector cross the road?
It wanted to be normal.
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