The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths and as shown, is where m, n, and p are positive integers, m and n are relatively prime, and p is not divisible by the square of any prime. what is m + n + p?

Set up the given triangle on x-y coordinates with right angle at (0,0). So the two vertices are at (5,0) and (0,2)

let (a,0) and (0,b) be two vertices of the equilateral triangle. So the third vertex must be at

for a pt (x,y) on line sx+ty=1, the minimum of 
equals to 

smallest value happens at 

so area is 

hence m=75, n=67, p=3
m+n+p = 75+67+3 = 145

1. 0.3 •4 is the same as 4 groups of 0.32. 7•0.8 is the same as 7 groups of 0.83. it increases by 0.5 every time, so 0.5x4 it increases by 0.9 every time, so 0.9x5. 0.9•12=10.86. 0.47•5=2.35i can finish the next ones in a separate post