27
Problem solving
Textbook: pp. 89–90
GoalKnow the support problems on area and apply them.
New words
support problem · tayanch masalaintersection point of the diagonals · diagonallar kesishgan nuqtacommon base · umumiy asosequal altitude · teng balandlik
Explanation
Support problems are ready-made results that are often used to solve other problems. 1) If the diagonals of a quadrilateral ABCD meet at O, then S(AOB) · S(COD) = S(BOC) · S(DOA). 2) In a trapezoid with bases AD and BC, S(AOB) = S(COD): triangles ABD and ACD have a common base and equal altitudes, and removing their common part AOD leaves equal areas. 3) A triangle with the same base and altitude as a parallelogram has half of the parallelogram’s area. General rule: triangles with equal bases and equal altitudes have equal areas.
Worked examples
In a convex quadrilateral S(AOB) = 6, S(BOC) = 4, S(COD) = 12. Then S(DOA) = 6 · 12 : 4 = 18.
A parallelogram has base 14 cm and altitude 6 cm, so its area is 84 cm². A triangle with that base and altitude has area 84 : 2 = 42 cm².
Class activity
“Equal areas”: draw a trapezoid on squared paper, split it with its diagonals into four triangles, then count and compare the two triangles next to the legs.
Practice
1
In a convex quadrilateral S(AOB) = 8, S(BOC) = 6, S(COD) = 12. Find S(DOA).
16
2
In a trapezoid with bases AD and BC, S(AOB) = 17. Find S(COD).
17
3
A parallelogram has area 90 cm². What is the area (cm²) of a triangle with the same base and altitude?
45
4
Why do the two triangles next to the legs of a trapezoid have equal areas?
Triangles ABD and ACD have common base AD and equal altitudes, so they have equal area; removing the common part AOD leaves AOB and COD with equal areas.