Question

A. How many non-degenerate scalene triangles with integer sides have a

perimeter of 27?


b. A square is inscribed in an equilateral triangle of side length 2 so that

two adjacent corners of the square lie on two sides of the triangle, and

the third side of the triangle contains one side of the square. The area of

the region inside the triangle and outside the square can be expressed as

a√b − c. What is a + b + c?

Answer 1

Hello,

a)18

Explanation:

`p=27, int[p/2]=13\\\\\begin{array}ca&p-a&b&c\\---&---&---&---\\1&26&13&13\\2&25&12&13\\3&24&12&12\\4&23&10&13\\&&11&12\\5&22&9&13\\&&10&12\\&&11&11\\6&21&8&13\\&&9&12\\&&10&11\\7&20&7&13\\&&8&12\\&&9&11\\&&10&10\\8&19&8&11\\&&9&10\\9&18&9&9\\---&---&---&---\\\end{array}`

number of triangle=18

b)

Side of the equilateral triangle =2

Height=√3 (using Pythagorean's theorem) (tr ACP)

Triangle ABJ has the same area as the triangle ABC

Uisng Thalès's theorem,

`(2-v)/(2)=(v)/(√(3) ) \\\\\\v=(2√(3) )/(2+√(3) ) \\\\\\v=(2√(3)*(2-√(3)) )/(1) } \\\\v=4√(3)-6\\\\v^2=84-48√(3) \\Area =a√(b)-c= √(3) -(84-48√(3) )=49√(3) -84\approx{0,87049...}\\\\So:\\b=3\\a=49\\c=84\\\\a+b+c=3+49+84=136\\`