By dividing a trapezium into two triangles show that its area is, half the sum of the parallel sides multiplied by the height (the same formula as the one given above).
Answer: Diagonal AC splits trapezium ABCD (AB = a ∥ DC = b, height h) into △ABC (base a, height h) and △ACD (base b, height h). Area = 12ah + 12bh = 12(a + b)h.
Step-by-step solution
Idea: Both triangles have the full height h, because the parallel sides are h apart.
- Draw the diagonal AC of trapezium ABCD, with AB = a, DC = b, AB ∥ DC and distance h between them.1 mark
- △ABC: base AB = a; its height is the distance from C to line AB = h. Area = 12ah. △ACD: base DC = b; height = distance from A to DC = h. Area = 12bh.1 mark
- Area of trapezium = 12ah + 12bh = 12(a + b)h.1 mark
Area = ½ah + ½bh = ½(a + b)h.
Answer to write in the exam
Join AC; AB = a ∥ DC = b, distance h
ar(△ABC) = 12ah, ar(△ACD) = 12bh
ar(ABCD) = 12ah + 12bh
∴ ar(ABCD) = 12(a + b)h
Common mistakes that cost marks
- Thinking △ABC has a smaller height. Its height (from C to line AB) is also h, measured outside the triangle.
- Using the slant side as a height.
How this can come in the exam
Short answer (2 marks)
A trapezium has parallel sides 7 cm and 13 cm, 6 cm apart. Find the areas of the two triangles made by a diagonal, and the area of the trapezium.
Show answer
12 × 7 × 6 = 21 cm2 and 12 × 13 × 6 = 39 cm2 (1 mark); total 60 cm2 = 12(20)(6) (1 mark).Try one yourself
A trapezium has parallel sides 5 cm and 11 cm and height 4 cm. Find the areas of the two triangles cut by a diagonal.
Show answer
10 cm2 and 22 cm2; total 32 cm2.
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