Area Theorems
ICSE Class 9 · Mathematics · Chapter 15
What this chapter covers
- Area, base and altitude
- Parallelograms on the same base and between the same parallels
- A triangle and a parallelogram on the same base
- Triangles on the same base and between the same parallels
- Triangles with equal areas on the same base
- Using the area theorems
- Common mistakes to avoid
Key points
- The area of a figure is the measure of the region it encloses; congruent figures have equal areas, but equal areas do not imply congruence.
- Figures between the same parallels have equal altitudes, because the distance between two parallel lines is the same everywhere.
- Theorem 1: parallelograms on the same base and between the same parallels are equal in area (proof by congruent triangles and subtraction from a common region).
- Every parallelogram therefore has the same area as the rectangle on its base between the same parallels: base times altitude.
- Theorem 2: a triangle is half of a parallelogram on the same base and between the same parallels, which proves ar(△) = tfrac12 × base × altitude.
- Theorem 3 (stated without proof): triangles on the same base and between the same parallels are equal in area.
- A median divides a triangle into two triangles of equal area.
- Theorem 4: triangles with equal areas on the same base, or on equal bases, have equal corresponding altitudes.
- Equal-area triangles on the same base and on the same side of it lie between the same parallels.
- In proofs, always write "ar" before equal areas and give the reason "on the same base and between the same parallels".
Questions answered
State the theorem about parallelograms on the same base and between the same parallels.
Parallelograms on the same base and between the same parallels are equal in area.
Define the altitude of a parallelogram.
The altitude of a parallelogram is the perpendicular distance between the chosen base and the side opposite to it.
Parallelogram PQRS has PQ = 11 cm and the altitude on PQ equal to 6 cm. Parallelogram PQTU lies on the same base between the same parallels. Find ar(PQTU).
ar(PQTU) = ar(PQRS) = 11 × 6 = 66 cm2, by Theorem 1.
Why does a slanted parallelogram not have a larger area than the rectangle on the same base between the same parallels, even though its slant sides are longer?
Area depends on the base and the perpendicular distance between the parallels, and both are the same for the two figures. The longer slant sides increase the perimeter, not the area.
ABCD is a parallelogram of area 90 cm2 and P is any point on CD. Find ar(△ APB).
By Theorem 2, ar(△ APB) = tfrac12 × 90 = 45 cm2.
In the proof of Theorem 2, why is the construction line drawn through B parallel to AP?
It creates a second parallelogram, ABQP, on the same base between the same parallels, so that Theorem 1 and the diagonal property can be combined.
AD is a median of △ ABC and ar(△ ABC) = 50 cm2. Find ar(△ ABD), giving a reason.
25 cm2. BD = DC and the two triangles have the same altitude from A, so the median halves the area.
ABCD is a trapezium with AB ∥ DC. Name two triangles on the base AB that have equal areas.
△ ABC and △ ABD, which are on the same base AB and between the same parallels AB and DC.
Past board exam questions
ICSE 2019 sample paper · 3 marks
In △ADE, B is a point on AD and C is a point on AE such that BC is parallel to DE. BE and CD are joined. Prove that: area △ABE = area △ACD
Show answer
△BCE and △BCD are on the same base BC and between the same parallels BC and DE.
∴ area △BCE = area △BCD.
Adding area △ABC to both sides:
area △ABC + area △BCE = area △ABC + area △BCD
Since C lies on AE and B lies on AD, this gives area △ABE = area △ACD. Hence proved.