Triangles and Quadrilaterals

1. Main Theorems 

 

TheoremStatementProof IdeaTheorem 1Parallelograms on same base & between same parallel lines have equal areaCongruent triangles by AASTheorem 2Triangle = ½ parallelogram on same base & same parallelsDiagonal divides parallelogram into 2 equal trianglesTheorem 3Triangles on same base & between same parallel lines have equal areaBoth = ½ same parallelogramMid-point Theorem (Extra important)Line joining mid-points of two sides → parallel to third side & half lengthVector or similar triangles

2. Area Formulas You Must Know

 

ShapeArea FormulaParallelogrambase × heightTriangle½ × base × heightTrapezium½ × (sum of parallel sides) × heightRhombusbase × height OR ½ × d₁ × d₂

3. Most Important Proof Questions (90% SEE)

1. Parallelograms on same base & same height → equal area
2. Triangle = ½ parallelogram on same base & height
3. Triangles on same base & same height → equal area
4. Mid-points E,F of AB,AC → EF || BC and EF = ½ BC
5. Varignon Theorem: Joining mid-points of quadrilateral → parallelogram
6. Area of parallelogram formed by mid-points = ½ original parallelogram

4. 5 Most Common SEE Questions + Short Solutions

1. ABCD parallelogram, E,F mid-points of AB,CD → prove EFGH parallelogram & area = ½ ABCD → EF || AD, FG || AB → EFGH parallelogram → area ½
2. Triangles with same base & same height → equal area → Both have same base and same perpendicular distance → area equal
3. Parallelogram ABCD, P on diagonal AC → prove area ΔABP = area ΔCDP → Same height from B and D to AC → equal area
4. Trapezium ABCD, AB||DC, E mid-point of AD, F mid-point of BC → prove EF || AB || DC & EF = ½(AB+DC) → Mid-line theorem in triangle
5. ABCD parallelogram, M mid-point of AB, N mid-point of AD → prove area AMND = ¼ ABCD → MN joins mid-points → MN = ½ CD → area ¼

visit this link for further practice

https://besidedegree.com/exam/s/academic