Parallelogram and Triangle on Same Base
Statement / Question: Prove that the area of a triangle is half of the area of a parallelogram (or parallelogram is double of triangle) standing on the same base and between the same parallel lines.
Given
△ABC & ▱ABEF on base AB between FC ∥ AB.
To Prove
Area(△ABC) = ½ Area(▱ABEF)
(OR Area(▱ABEF) = 2 × Area(△ABC))
Construction
Draw MN ⊥ base AB.
Theoretical Proof
| Statements | Reasons |
|---|---|
| 1. Area of △ABC = ½ × AB × MN | 1. Area of triangle = ½ × Base × Height |
| 2. Area of ▱ABEF = AB × MN | 2. Area of parallelogram = Base × Height |
|
3. Area of △ABC = ½ Area of ▱ABEF
(OR: Area of ▱ABEF = 2 × Area of △ABC) |
3. From Statements (1) and (2) |
Triangle and Rectangle on Same Base
Statement / Question: Prove that the area of a triangle is half of the area of a rectangle (or rectangle is double of triangle) standing on the same base and between the same parallel lines.
Given
△ABC & ▭ABEF on base AB between FC ∥ AB.
To Prove
Area(△ABC) = ½ Area(▭ABEF)
(OR Area(▭ABEF) = 2 × Area(△ABC))
Construction
In ▭ABEF, side AF ⊥ AB.
Theoretical Proof
| Statements | Reasons |
|---|---|
| 1. Area of △ABC = ½ × AB × AF | 1. Area of triangle = ½ × Base × Height |
| 2. Area of rectangle ▭ABEF = AB × AF | 2. Area of rectangle = Length × Breadth |
|
3. Area of △ABC = ½ Area of rectangle ▭ABEF
(OR: Area of rectangle ▭ABEF = 2 × Area of △ABC) |
3. From Statements (1) and (2) |