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A triangle has consecutive side and height lengths for some integer n, as shown in image. What is its perimeter?
By solving the equation
n^2 + (n+2-x)^2 = (n+3)^2 ---- 1 &
n^2 + x^2 = (n+1)^2 ---- 2
We can get n = 12 and x = 5 as real solutions.
Now if n = 12 and the base of the triangle = 14 is divided as 5 and 9 between the 2 right angled inner triangles.
ie., The Pythagoras applied to first triangle is 12^2 + 5^2 = 13^2 = 144 + 25 = 169
and for the second triangle is 12^2 + 9^2 = 15^2 = 144 + 81 = 225
An acute triangle has a circumcircle, as shown in image. Minor arcs are reflected about each side of the triangle.
Do the three reflected arcs intersect at the same point?
Note: Share your reasoning...
A rectangle with integer side lengths is cut into 12 squares of lengths 2,2,3,3,5,5,7,7,8,8,9,9. What is the perimeter of this rectangle?
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