Area Theorems
ICSE · Class 9 · Mathematics
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Two parallelograms ABCD and ABEF are on the same base AB and between the same parallels. If the area of parallelogram ABCD is 48 cm², what is the area of parallelogram ABEF?
A triangle ABC and parallelogram ABDE are on the same base AB and between the same parallels. If the area of parallelogram ABDE is 60 cm², find the area of triangle ABC.
Two triangles PQR and PQS are on the same base PQ and between the same parallels. If the area of triangle PQR is 25 cm², what is the area of triangle PQS?
In triangle ABC, D is the midpoint of BC. What is the ratio of areas of triangles ABD and ADC?
Sample Questions
A parallelogram has area 72 cm². A triangle is drawn on the same base and between the same parallels as the parallelogram. Find the area of the triangle.
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36 cm²
Area of triangle = ½ × Area of parallelogram (when on same base and between same parallels). Area of triangle = ½ × 72 = 36 cm².
Which of the following statements about area theorems are correct?
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Parallelograms on same base and between same parallels are equal in area, Triangle area is half of parallelogram area on same base, Triangles on same base and between same parallels are equal in area
The first, second, and fourth statements are fundamental area theorems. The third statement is false as triangle areas depend on base and height. The fifth statement is false as areas depend on dimensions, not shape type.
In triangle PQR, S divides side QR in the ratio 3:2. If the area of triangle PQR is 50 cm², find the area of triangle PQS.
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30 cm²
QS:QR = 3:5 (since QS:SR = 3:2, so QS:QR = 3:(3+2) = 3:5). Area of triangle PQS/Area of triangle PQR = QS/QR = 3/5. Area of triangle PQS = (3/5) × 50 = 30 cm².
Two parallelograms have equal bases of 8 cm each and are between the same parallels with height 6 cm. What can you conclude about their areas?
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Areas are equal
Parallelograms on equal bases and between same parallels are equal in area. Both have area = base × height = 8 × 6 = 48 cm².
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