In ICSE Class 9 Mathematics, the topic of “Theorems on Area” typically includes several important theorems related to the area of triangles, quadrilaterals, and circles. Here’s a summary of the key theorems covered in this topic:
1. **Triangle Area Theorems**:
– **Area of a Triangle**:
– The area of a triangle is given by the formula: Area = 1/2 * base * height.
– **Heron’s Formula**:
– For a triangle with sides of lengths a, b, and c, the area is given by:
Area = √[s(s – a)(s – b)(s – c)], where s = (a + b + c) / 2 (semiperimeter).
2. **Quadrilateral Area Theorems**:
– **Area of a Trapezium**:
– The area of a trapezium is given by the formula: Area = 1/2 * (sum of parallel sides) * height.
– **Midpoint Theorem**:
– If a line segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and equal to half of it.
3. **Circle Area Theorems**:
– **Area of a Circle**:
– The area of a circle is given by the formula: Area = πr², where r is the radius of the circle.
– **Sector of a Circle**:
– The area of a sector of a circle is a fraction of the area of the whole circle, given by: Area of Sector = (θ/360) * πr², where θ is the central angle of the sector.
4. **Segment of a Circle**:
– The area of a segment of a circle is the area of the sector minus the area of the triangle formed by the segment’s chord and two radii.
These theorems are fundamental in geometry and are used to calculate the areas of various shapes in different contexts. Understanding and applying these theorems is essential for solving geometry problems involving areas in the ICSE Class 9 Mathematics curriculum.