Find a representation as a product of \(a^n - b^n\) for general \(a,b,n\).
Let \(a,b,c,d\) be positive real numbers. Prove that \((a+b)\times(c+d) = ac+ad+bc+bd\). Find both algebraic solution and geometric interpretation.
Let \(a,b,c,d\) be positive real numbers such that \(a\geq b\) and \(c\geq d\). Prove that \((a-b)\times(c-d) = ac-ad-bc+bd\). Find both algebraic solution and geometric interpretation.
Using the area of a rectangle prove that \(a\times b=b\times a\).
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a square, or slightly harder in a shape of a given rectangle.
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a hexagon
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a given triangle.
Draw how Robinson Crusoe should put pegs and ropes to tie his goat in order for the goat to graze grass in the shape of a shape like this
Generally, when a line intersects a circle, it creates two different points of intersection. However, sometimes there is only one point. In such case we say the line is tangent to the circle. For example on the picture below the line \(CD\) intersects the circle at two points \(D\) and \(E\) and the line \(CB\) is tangent to the circle. To solve the problems today we will need the following theorem.
Theorem: The radius \(AB\) is perpendicular to the tangent line \(BC\).
Sometimes one can guess certain multiples of a number just by looking at it, the idea of this sheet is to learn to recognise quickly using tricks when a natural number is divisible by another number.