Area of triangle DOA?
A hard-level "Area" problem.Find a ratio once, then use it again somewhere else in the same figure. That is the standard way through a two-step area-ratio problem.
GivenThe diagonals of quadrilateral ABCD meet at O. Triangle AOB = 6 cm², BOC = 9 cm², COD = 12 cm².
Hints
- There is no direct route to "?". Instead, find out what ratio O splits the diagonal AC into. That ratio will carry us all the way there.
- Look at the top pair, AOB and BOC. Both have apex B, and their bases AO and OC lie on one straight line — so they have the same height.
- Same height means area ratio = base ratio. So AO : OC = 6 : 9 = 2 : 3.
- This is the turning point: the ratio AO : OC works just as well on the bottom pair. Triangles DOA and COD both have apex D, with the same bases AO and OC — same height again.
- So ? : 12 = 2 : 3. Since 12 is 3 parts, one part is 4, and "?" is 2 of them.
- Looking back, the products of the facing pairs are equal: 6×12 = 9×8. It holds for any quadrilateral.
Answer 8cm²
More "Area" problems
- Area of the inner triangle?
- Area of quadrilateral ABCD?
- Area of the overlap?
- Total area of the two crescents?
- Area swept by the square?
- What area of the table is covered by paper?