SOLUTION: The area of a rectangle is 45, and its perimeter is 28. Find its dimensions and diagonal.

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Question 1125600: The area of a rectangle is 45, and its perimeter is 28. Find its dimensions and diagonal.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849)   (Show Source):
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let dimensions be and

given:
The area of a rectangle is , so we have
....solve or
......eq.1

its perimeter is , so we have

...solve or
...eq.2

from eq.1 and eq.2 we have

...solve or

solutions:
->
or
->

go to eq.2, substitute
...eq.2

so, we will choose the length to be and the width to be -> dimensions

and we can ind diagonal using Pythagorean theorem since diagonal cuts rectangle into two right angle triangles where diagonal is actually hypotenuse

approximately

Answer by ikleyn(52780)   (Show Source):
You can put this solution on YOUR website!
.

            This problem admits easy MENTAL solution.

Since the perimeter is 28 units, the sum of the length and the width is half ot it, i.e. 14 units. Hence, the average of the length and the width is half of 14, i.e. 7 units. The average is equally remoted value from the length and the width, i.e. L = 7 + u, W = 7 - u, where "u" is that common distance. Then the area 45 = L*W = (7+u)*(7-u) = 49 - u^2, which means u^2 = 49 - 45 = 4 and hence u = = 2 units. Then L = 7 + 2 = 9 and W = 7 - 2 = 5. Then the diagonal is = = .

Solved.