ABCD is a trapezium with parallel sides AD = 3 cm and BC = 5 cm. E and F are the midpoints of the non-parallel sides. Find the ratio of the areas of the 4-gons AEFD and EBCF.
Step-by-step solution
To find: ar(AEFD) : ar(EBCF)
Idea: The segment joining the midpoints of the slanting sides of a trapezium is parallel to the parallel sides and equal to half their sum. It also lies exactly halfway up, so both smaller trapeziums have the same height.
- E and F are midpoints of the non-parallel sides, so EF ‖ AD ‖ BC and EF = ½(AD + BC) = ½(3 + 5) = 4 cm (draw diagonal AC with midpoint G: in ∆ABC, EG ‖ BC and EG = ½BC; in ∆ACD, GF ‖ AD and GF = ½AD; both EG and GF are parallel to BC through G, so E, G, F are collinear and EF = EG + GF).1½ marks
- Let h be the height of ABCD. Draw AY ⟂ BC, meeting EF at X. In ∆ABY, E is the midpoint of AB and EX ‖ BY, so X is the midpoint of AY: AX = XY = h/2. Both parts have height h/2.1 mark
- ar(AEFD) = ½(3 + 4) × h2 = 7h4; ar(EBCF) = ½(4 + 5) × h2 = 9h4.1 mark
- Ratio = 7h4 : 9h4 = 7 : 9.½ mark
Check: Take h = 4: whole area = ½(3 + 5) × 4 = 16; parts = ½(3 + 4) × 2 = 7 and ½(4 + 5) × 2 = 9; 7 + 9 = 16 ✓.
Answer to write in the exam
EF ‖ AD ‖ BC and EF = ½(AD + BC) = ½(3 + 5) = 4 cm
AY ⟂ BC meets EF at X; E midpoint of AB, EX ‖ BY ⇒ AX = XY = h/2
ar(AEFD) = ½(3 + 4)(h/2) = 7h/4; ar(EBCF) = ½(4 + 5)(h/2) = 9h/4
∴ ar(AEFD) : ar(EBCF) = 7 : 9
Common mistakes that cost marks
- Answering 1 : 1 because EF cuts the height in half. The parallel sides of the two parts are different (3, 4 and 4, 5).
- Taking EF = ½(5 − 3) = 1 cm. The midline is half the sum, not half the difference.
- Answering 3 : 5 (the ratio of the parallel sides).
How this can come in the exam
The parallel sides of a trapezium are 2 cm and 10 cm. The segment joining the midpoints of the other two sides divides its area in the ratio
- 1 : 5
- 1 : 2
- 3 : 5
- 1 : 1
Show answer
(B) 1 : 2
Midline = ½(2 + 10) = 6 cm. Both parts have the same height, so the areas are in the ratio (2 + 6) : (6 + 10) = 8 : 16 = 1 : 2.
Try one yourself
A trapezium has parallel sides 4 cm and 12 cm. Find the ratio in which the segment joining the midpoints of its non-parallel sides divides its area.
Show answer
Midline = 8 cm; ratio = (4 + 8) : (8 + 12) = 12 : 20 = 3 : 5.
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