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Mathematics Class 9 - Areas Of-Parallelograms-And-Triangles Notes

Comprehensive study notes for Class 9 - Areas Of-Parallelograms-And-Triangles olympiad preparation

# Areas of Parallelograms and Triangles

In this chapter, you will learn how to find the area of parallelograms and triangles, understand their properties, and solve problems using formulas. By the end of this chapter, you will be able to apply these concepts to real-life situations and geometry problems.

## Key Concepts
  • Area: The amount of surface covered by a shape.
  • Parallelogram: A quadrilateral with opposite sides parallel and equal.
  • Triangle: A polygon with three sides and three angles.
## Area of a Parallelogram

The area of a parallelogram is given by:

Area = Base × Height

  • Base: Any side of the parallelogram.
  • Height: The perpendicular distance from the base to the opposite side.

Example: If the base is 8 cm and the height is 5 cm,
Area = 8 × 5 = 40 cm²

## Area of a Triangle

The area of a triangle is given by:

Area = ½ × Base × Height

  • Base: Any side of the triangle.
  • Height: The perpendicular distance from the base to the opposite vertex.

Example: If the base is 10 cm and the height is 6 cm,
Area = ½ × 10 × 6 = 30 cm²

## Properties and Applications
  • Parallelograms with the same base and height have equal areas.
  • Triangles with the same base and height have equal areas.
  • Area helps in finding the amount of material needed to cover a surface.
## Practice Questions
  1. Find the area of a parallelogram with base 12 cm and height 7 cm.
  2. Calculate the area of a triangle with base 15 cm and height 8 cm.
  3. If two parallelograms have the same base and height, will their areas be equal?
  4. A triangle has an area of 24 cm² and a base of 8 cm. What is its height?
  5. Why is the area of a triangle half the area of a parallelogram with the same base and height?
## Challenge Yourself
  • Draw a parallelogram and a triangle with the same base and height. Calculate and compare their areas.
  • Find the area of a triangle if its sides are 7 cm, 24 cm, and 25 cm (use Heron's formula).
## Did You Know?
  • The formula for the area of a triangle can be used for all types of triangles.
  • Heron's formula is used to find the area of a triangle when all sides are known.
## Glossary
  • Base: The side of a shape used for measurement.
  • Height: The perpendicular distance from the base to the opposite side or vertex.
  • Heron's Formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 and a, b, c are the sides of the triangle.
## Answers to Practice Questions
  1. Area = 12 × 7 = 84 cm²
  2. Area = ½ × 15 × 8 = 60 cm²
  3. Yes, their areas will be equal.
  4. Height = (Area × 2) / Base = (24 × 2) / 8 = 6 cm
  5. Because a triangle is half of a parallelogram with the same base and height.

Practice finding areas to master geometry and solve real-life problems!

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