Class-9-Heron-s-Formula and Quadrilaterals for class IX.pptx

Class IX area of Triangles with different formulas

1. CHAPTER - 12 HERON’S FORMULA By- Sujeet Kumar PGT MATHEMATICS

2. AREA OF TRIANGLE BY HERON’S FORMULA Heron was born in about 10AD possibly in Alexandria in Egypt. His works on mathematical and physical subjects are so numerous and varied that he is considered to be an encyclopedic writer in these fields. His geometrical works deal largely with problems on mensuration. He has derived the famous formula for the area of a triangle in terms of its three sides. HERON (10AD - 75AD)

3. INTRODUCTION In earlier classes we have studied to find an area and perimeter of a triangle Perimeter is sum of all sides of the given triangle Area is equal to the total portion covered in a triangle

4. Area of triangle = ½ x base x height Perimeter = sum of all sides of triangle 8 cm Perimeter = sum of all sides = 5 + 5 + 8 = 18 cm 6 cm Area = ½ x base x height Area = ½ x 8 x 6 Area = 24 cm 2

5. TYPES OF TRIANGLES EQUILATERAL TRIANGLE ISOSCELES TRIANGLE RIGHT ANGLE TRIANGLE SCALENE TRIANGLE

6. AREA OF RIGHT ANGLE TRIANGLE 8 cm 5 cm In a right angle triangle we can directly apply the formula to find the area of the triangle, as two sides containing the right angle as base and height. Consider the following figure – Base = 5 cm Height = 8 cm Area = ½ x 8 x 5 = 20 cm 2

7. AREA OF EQUILATERAL TRIANGLE 5 cm Find the area of an equilateral triangle with side10 cm. Here, we can find height by pythagoras theorem So here height = √ 75 = 5√3 Area = ½ x base x height = ½ x 10 x 5√3 = 25√3 cm 2

8. AREA OF ISOSCELES TRIANGLES 8 cm 4 cm find out the area of an isosceles triangle whose 2 equal sides are 5 cm and the unequal side is 8 cm Here height can be find by pythagoras theorem So, h = 3 cm Area = ½ x base x height = ½ x 8 x 3 = 12 cm 2

9. HERON’S FORMULA Area of triangle = √s(s- a)(s-b)(s- c) Where a , b and c are the sides of the triangle , and s = semi perimeter, i.e., half of perimeter of the triangle = a + b + c 2

10. IMPORTANCE OF HERON’S FORMULA This formula is helpful where it is not possible to find height of the triangle easily. It is also helpful in finding area of quadrilaterals.

11. Q- Find the area of triangle whose sides are 3cm, 4cm & 5 cm respectively. Area of triangle = √s(s- a)(s- b)(s- c) = 3+4+5 = 6 2 Area of triangle = √6(6-3)(6-4)(6- 5) = √ 6 x 3 x 2 x 1 = 6 cm² As s = a + b + c 2