The parallel sides of a trapezium are a cm and b cm long, where a and b are integers, a < b.
The trapezium is split into two smaller trapezia of equal area by a line of length c cm which is parallel to the sides of length a cm and b cm.
Given that c is also an integer, what is the smallest possible value of c?
Let the height of the largest trapezoid = h.
The line which splits this must split the height into proportionate parts. The height connecting bases C and A is (c-a)/(b-a)*h
Thus we have the formula relating the large trap to one of the pieces:
1/2*h*(b+a) = 2*1/2*(c-a)/(b-a)*h*(c+a)
(b+a)(b-a) = 2(c+a)(c-a)
b^2+a^2=2c^2
The smallest positive integer solution with a<b is (1,7,5)
So c=5
Note: (5-1)/(7-1) = 4/6 = 2/3 so if the heights of the trapezoids must also be integers, h=3 and the smaller heights are 1 and 2.
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Posted by Jer
on 2024-07-18 10:28:26 |
Solution
