| --- |
| source: ncert |
| topic: geometry |
| class_level: class_9 |
| chapter: ch9 |
| difficulty: intermediate |
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| |
| 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). |
