Sweden Math Olympiad Geometry Problem | 2 Different Methods

[@jimlocke9320]

At 16:11, 1/(tan(37.5°) is given an approximate value of 1.3032. However, there is an exact value. There is a trigonometric identity tan(Θ/2) = (1 - cos(Θ))/sin(Θ). Here, Θ = 75° and the exact values for sin(75°) and cos(75°), from ratio of sides of the 15°-75°-90° right triangle, are: sin(75°) = (√3 + 1)/2√2 and cos(75°) = (√3 - 1)/2√2). So, tan(37.5°) = (1 - (√3 - 1)/2√2))/((√3 + 1)/2√2) =( 2√2 - (√3 - 1))/(√3 + 1) which, after multiplying numerator and denominator by (√3 - 1), further simplifies to √6 - √2 + √3 - 2. So, r = 2/(√3 + 1/(√6 - √2 + √3 - 2)), which should simplify to r = (1 -√2 +√3)/2.

[@souzasilva5471]

I learned that the area of ​​a triangle is equal to the semiperimeter times the apothem, not times the radius.

[@jimlocke9320]

The apothem is defined as a line from the center of a regular polygon at right angles to any of its sides. The equilateral triangle is the only regular 3 sided polygon. It is clear that a circle inscribed in an equilateral triangle will have a radius equal to the apothem. Unless the triangle is equilateral, it will not have an apothem, but it will be possible to inscribe a circle in any triangle.

[@professorrogeriocesar]

Muito bom. Usei a Lei dos Senos e as fórmulas de área: A=pr, A=ab*sin(alpha)/2.

[@superilior]

Same

[@五十嵐特許事務所]

r=2/(root(6)+2-root(2))

[@MahdiSheikhAadan]

Please we need video math hunter another video calculus booster please we watch this chanel we have no video 2 days

[@thomasleach9417]

R= .496