Problem 27 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
arearatio
A triangle ABC is divided into four parts by two straight lines, as shown. The areas of the smaller triangles are 1, 3 and 3. What is the area of the original triangle?
Show answer
Answer: A — 12
Show hints
Hint 1 of 2
Triangles sharing the same height have areas in the ratio of their bases.
Still stuck? Show hint 2 →
Hint 2 of 2
Use the two equal-3 areas to find a base ratio, then chase the areas up to the whole triangle.
Show solution
Approach: use base ratios from shared-height triangles
- The cevians cut ABC into triangles of areas 1, 3, 3 and one quadrilateral.
- Comparing triangles on a common base/height, the segment ratios force the quadrilateral's area to be 5.
- Total area = 1 + 3 + 3 + 5 = 12.
Mark:
· log in to save