### Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

### Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

# Same Height

##### Stage: 4 Challenge Level:

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b , what is the area of the trapezium?

The following solution was done by Ling Xiang Ning, Allan from Tao Nan School, Singapore

First note that triangles SPR and SQR are equal in area (same base and height) so triangles SPT and RQT are equal in area; suppose this area is c .

Now triangles SPT and TPQ have the same height (with their common base on SQ) and the ratio of their areas is:
$$\frac{c}{b} = \frac{\text{Area}(SPT)}{\text{Area}(TPQ)} = \frac{ST}{TQ}$$
$$\frac{a}{c} = \frac{\text{Area}(SRT)}{\text{Area}(TRQ)} = \frac{ST}{TQ}$$
hence $c/b = a/c$.

Then $c^2 = ab$ and $c = \sqrt{ab}$

Therefore, the total area of the trapezium is $a + b + 2\sqrt{ab}$.