The heights drawn from the apex of the obtuse angle of the parallelogram are 2: 4.

The heights drawn from the apex of the obtuse angle of the parallelogram are 2: 4. What is the smaller side of the parallelogram if the perimeter is 90 cm?

Since the parallelogram has the same opposite angles, the angle BAD = BCD.

Consider right-angled triangles ABН and BСК, in which the angles H and K are straight, and the angles BAN and BSC are equal, then the triangle BAН is similar to the triangle BCК in an acute angle.

By condition, BH / BK = 2/4, then AB / BC = 2/4.

The perimeter by condition is 90 cm, and since the opposite sides are equal, AB + BC = 90/2 = 45 cm. AB = 45 – BC.

Then: (45 – BC) / BC = 2/4.

2 * BC = 4 * (45 – BC).

2 * BC + 4 * BC = 180.

6 * BC = 180.

BC = 180/6 = 30 cm.

Then AB = 45 – BC = 45 – 30 = 15 cm.

Answer: The length of the shorter side is 15 cm.