Rectangle abcd5/7/2023 ![]() Oddly enough taking even values $m-n=2, r-s=4,r s=6,m n=12$ so $r=5 s=1 n=5 m=7$ gives us a smaller integer $7^2 1=5^2 5^2=50$ and as smaller sum of $18$ but the terms are not distinct.) (I'm not sure how to verify this other than trial and error. This would be the smallest set of four distinct values for $m-n,r-s,r s,m n$. Now $m n, m-n$ must be the same parity and and we could do $r s=5, r-s=3$ and $m n=15, m-n=1$ so $m=8 n=7 r=4 s=1$. We must have $m^2 s^2 = n^2 r^2$ and that simply requires finding the smallest integer that is a sum of two distinct perfect squares in two possible ways that will have the smallest sum. Place it on a coordinate system so that the $E$ is on $(0,0)$ and $A = (-a, b) B= (-a,-c), C=(d,b) D=(d,-c)$ and we need for $\sqrt=s$ are all integers. Simplifying this equation we get the equation. With this information, we can write the equation Doing some more angle chasing, we can find that, as they both share and they both have a right angle. Let point be the point on line so that lines and are perpendicular, and we get that and. Through simplifying this equation, we get that. Using this information, we can write the equation. We find that through some angle chasing (they both have a right angle, and they both share angle. We find that (through Pythagorean Theorem),, and. Solution 7 (system of equations through angle similarity)įirst, using given information, we can find the values of some line segments in the figure. Plugging this into our equation for line gives us, so
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