A rectangle with integer sides has a diagonal stripe which starts 1 unit from the diagonal corners, as in the diagram. The area of the stripe is exactly half of the area of the rectangle. What is the perimeter of this rectangle?
Answers
Given : A rectangle with integer sides has a diagonal stripe which starts 1 unit from the diagonal corners, as in the diagram.
The area of the stripe is exactly half of the area of the rectangle.
To Find : the perimeter of this rectangle
Solution:
Let say rectangle dimensions are a and b
Hence area = ab sq units
Area of strip is half hence area of remaining two triangles together is also half
= ab/2
area of two triangles = 2 * (1/2) * (a - 1) ( b - 1)
= ab - a - b + 1
ab - a - b + 1 = ab/2
=> ab/2 + 1 = a + b
=> ab/2 - a = b - 1
=> ab - 2a = 2b - 2
=> a(b - 2) = 2b - 2
=> a = (2b - 2)/(b - 2)
taking b - 2 = 1 as a and b are integers
=> b = 3 and a = 4
( 4 , 3) satisfy this
as then ( 4 * 3)/2 + 1 = 4 + 3
=> 6 + 1 = 7
=> 7 = 7
Hence perimeter of rectangles = 2 ( 4 + 3) = 14 unit
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