Using the area formula backwards

A triangle has three bases and three matching heights, and all three pairs give the same area (it is one triangle, so it has one area). That fact is a tool: work out the area the easy way, then use it to find a height you could not measure.

In ∆ABC, the altitude AX to BC is 3 units and BC is 5 units. So Area = ½ × AX × BC = ½ × 3 × 5 = 15/2 square units.

Now take AC as the base instead. AC is 4 units and BY is the altitude to it, so Area = ½ × BY × AC = ½ × 4 × BY = 2BY.

Both are the same area, so 2BY = 15/2, which gives BY = 15/4 = 3.75 units. A length nobody measured, found through the area.

Second question. Draw both diagonals of a rectangle. They cut it into four triangles. Are the four areas equal?

Careful: the four triangles are clearly not congruent. Two are short and wide, two are tall and narrow. Equal area and congruence are different things.

Compare two triangles next to each other, 1 and 2. Take OD and OB as their bases. Both triangles share the same top point, so they have the same height to that line. And OB = OD, because the diagonals of a rectangle cut each other in half. Equal bases, equal heights, so equal areas.

Go round the figure the same way and all four are equal. This argument did not really need a rectangle, which gives a general rule worth remembering: in any triangle, the line from a corner to the midpoint of the opposite side splits it into two triangles of equal area.