source: ncert
topic: geometry
class_level: class_9
chapter: ch9
difficulty: intermediate
Class 9 | Ch9: Areas of Parallelograms and Triangles KEY CONCEPT: Two figures with equal areas are not necessarily congruent. Two congruent figures always have equal areas. THEOREMS: Theorem 1: Parallelograms on same base and between same parallels are equal in area. Theorem 2: Triangles on same base and between same parallels are equal in area. Theorem 3: Median divides triangle into two equal area triangles. Theorem 4: Triangle area = half of parallelogram area if same base and same parallels. AREA FORMULAS: Parallelogram = base × height (perpendicular height). Triangle = (1/2) × base × height. If parallelogram ABCD and triangle ABE are on same base AB between same parallels: Area(ABE) = (1/2) × Area(ABCD). SOLVED EXAMPLES: Example 1: ABCD is parallelogram. E is point on DC. Show area(ABE)=area(ABD). Both triangles ABD and ABE have same base AB. Both between same parallels AB and DC. Area(ABD)=Area(ABE)=(1/2)Area(ABCD). Example 2: Parallelogram ABCD and rectangle ABEF on same base AB between same parallels. Area(ABCD)=Area(ABEF)=AB×h. Example 3: In triangle PQR, S is midpoint of QR. Area(PQS)=Area(PRS). PS is median. Median divides triangle into two equal areas. Example 4: Parallelogram base=8cm, height=5cm. Area=8×5=40cm^2. Triangle between same base and parallels: area=20cm^2. COMMON MISTAKES: Height must be PERPENDICULAR to base. Two parallelograms on same base with same height have equal area even if they look different. Median divides area in HALF (not sides in half — it divides opposite side in half).