Mathematics Notes for Class 11 Areas Of-Parallelograms-And-Triangles

Areas of Parallelograms and Triangles

In this chapter, you will explore advanced concepts related to the areas of parallelograms and triangles. You will learn about their properties, important theorems, proofs, and applications in coordinate geometry. By the end of this chapter, you will be able to solve complex problems involving these shapes and understand their significance in mathematics.

Key Concepts

Parallelogram: A quadrilateral with both pairs of opposite sides parallel and equal.
Triangle: A polygon with three sides and three angles.
Area: The measure of the region enclosed by a figure.

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.

Properties:

Parallelograms on the same base and between the same parallels are equal in area.
If a parallelogram and a rectangle are on the same base and between the same parallels, their areas are equal.

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.

Properties:

Triangles on the same base and between the same parallels are equal in area.
The area of a triangle is half the area of a parallelogram on the same base and between the same parallels.

Area in Coordinate Geometry

The area of a triangle whose vertices are A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is:

Area = ½ | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |

This formula is useful for finding the area of a triangle when the coordinates of its vertices are known.

Important Theorems

Theorem 1: Parallelograms on the same base and between the same parallels are equal in area.
Theorem 2: Triangles on the same base and between the same parallels are equal in area.
Theorem 3: The area of a triangle is half the area of a parallelogram on the same base and between the same parallels.

Applications

Finding areas of land plots and geometric figures.
Solving problems in coordinate geometry.
Proving geometric properties using area relations.

Practice Questions

Prove that parallelograms on the same base and between the same parallels are equal in area.
Find the area of a triangle with vertices at (2, 3), (4, 8), and (5, 3).
If the area of a parallelogram is 60 cm² and its base is 12 cm, find its height.
Show that the area of a triangle is half the area of a parallelogram on the same base and between the same parallels.
Given two triangles on the same base and between the same parallels, explain why their areas are equal.

Challenge Yourself

Given points A(1, 2), B(4, 6), and C(7, 2), find the area of triangle ABC using the coordinate formula.
Draw two parallelograms on the same base and between the same parallels. Calculate and compare their areas.
If the area of a triangle is 24 cm² and its base is 8 cm, find its height.

Did You Know?

The area formula for a triangle works for all types of triangles, including scalene, isosceles, and equilateral.
Heron's formula can be used to find the area of a triangle when all three 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.
Coordinate Geometry: The study of geometric figures using the coordinate plane.
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

By drawing two parallelograms on the same base and between the same parallels and showing their areas are equal using the area formula.
Area = ½ |2(8-3) + 4(3-3) + 5(3-8)| = ½ |2×5 + 4×0 + 5×(-5)| = ½ |10 + 0 - 25| = ½ × 15 = 7.5 square units.
Height = Area / Base = 60 / 12 = 5 cm.
Because a triangle is exactly half of a parallelogram on the same base and between the same parallels.
Because they have the same base and height, so their areas are equal by the area formula.

Mastering areas of parallelograms and triangles is key to solving many geometry and real-life problems!

Mathematics Class 11 - Areas Of-Parallelograms-And-Triangles Notes