Problem 30 · 2021 Math Kangaroo
Stretch
Geometry & Measurement
arearatio
The diagram shows a quadrilateral divided into 4 smaller quadrilaterals with a common vertex K. The other labelled points divide the sides of the large quadrilateral into three equal parts. The numbers indicate the areas of the corresponding small quadrilaterals. What is the area of the shaded quadrilateral?
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Answer: C — 6
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Hint 1 of 2
Because the side-points trisect the big quadrilateral's sides, triangles sharing the vertex K have areas in fixed ratios.
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Hint 2 of 2
Set up the proportional relations among the four corner quadrilaterals and solve for the shaded one.
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Approach: use the equal-trisection area ratios
- The labelled points trisect each side of the big quadrilateral, so triangles that share the vertex K and sit on equal side-thirds have equal areas.
- Splitting each of the four small quadrilaterals through K into two triangles and matching the equal-base pairs ties the three known areas 8, 10 and 18 to the shaded one.
- Solving those balance relations gives the shaded quadrilateral an area of 6.
- So the answer is C.
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