Problems

A square has been divided into \(4\) rectangles and a square. If the rectangle in the bottom left corner has dimensions \(1 \times 4\) and the one in the top right is \(2 \times 5\), what is the area of the small square in the middle?

On the diagram each number denotes the area of a rectangle it is written into. What are the areas of the other rectangles?

Draw any quadrilateral with all sides of different length and divide it into \(5\) polygons of equal area.

Line \(AB\) is parallel to line \(CD\) and line \(AD\) is parallel to line \(BE\). Show that triangles \(ADE\) and \(ABC\) have equal areas.

A quadrilateral \(ABCD\) is given. Points K and L belong to the side \(AB\) and \(AK=KL=LB\) and points \(N\) and \(M\) belong to the side \(CD\) and \(CM=MN=ND\). Show that the area of the quadrilateral \(KLMN\) is \(\frac13\) of the area of the quadrilateral \(ABCD\).

A quadrilateral \(ABCD\) is given. Point \(M\) is a midpoint of \(AB\) and point \(N\) is a midpoint of \(CD\). Point \(P\) is where segments \(AN\) and \(DM\) meet, point \(Q\) is where segments \(MC\) and \(NB\) meet. Show that the sum of areas of triangles \(APD\) and \(BCQ\) is equal to the area of the quadrilateral \(MQNP\).

A square was cut with two parallel lines that are \(6\) cm apart. One of them goes through the top right corner and the other through the bottom left corner. The three regions obtained this way, two triangles and a parallelogram, have equal areas. What is the area of the square?

The area of the triangle \(\triangle AEC\) is \(4\), the area of the triangle \(\triangle BCE\) is \(9\), the area of the triangle \(\triangle ABC\) is \(21\). What is the area of the triangle \(\triangle ADE\)?

A series of squares are connected by touching vertices. Some triangles were drawn outside and inside of the chain. Show that the total green area is the same as the total red area.

In a triangle \(\triangle ABC\), \(D\) is the midpoint of \(BC\), and \(E\) is the midpoint of \(AD\). \(F\) is the intersection of the side \(AC\) with \(BE\). What is the area of the triangle \(\triangle AEF\) as a proportion of the area of the triangle \(\triangle ABC\)?