Problem 10 · 2024 Math Kangaroo
Medium
Geometry & Measurement
area-decomposition
The diagram shows a rhombus. Its area is increased by adding two right-angled triangles (see diagram), where \(\angle NMA = \angle BMN = 90^\circ\). By what percentage does this increase the area?
Show answer
Answer: E — 50%
Show hints
Hint 1 of 2
Split the rhombus by a diagonal into two equal triangles and compare an added triangle to one of them.
Still stuck? Show hint 2 →
Hint 2 of 2
With the given right angles, each added triangle has the same base and height as half the rhombus.
Show solution
Approach: compare each added triangle's area to half the rhombus
- The horizontal diagonal splits the rhombus into two congruent triangles, so each half is half the rhombus.
- The right angles at N make the two added triangles together fit exactly onto the lower half-triangle of the rhombus, so they add up to half the rhombus.
- Adding half the rhombus increases the area by 50%.
Mark:
· log in to save