Area Theorems for Triangles and Parallelograms with Proofs and Solved Questions
The area of the theorem has basically few properties which are as follows:
A parallelogram is divided into two triangles of equal areas by the diagonal.
The ratio of the areas of 2 triangles with a similar height is equivalent to the ratio of their bases.
The ratio of the areas of 2 triangles on the same base is equivalent to the ratio of their heights.
The area of triangles that are congruent is equal.
Area Theorems
The theorems state some link between the areas of these geometric objects under the condition when they lie between the same parallel lines and on the same base (or equal bases). Following are the area theorem axioms:
Theorem 1
Diagonal of a parallelogram cut it half into 2 triangles of the same area. Parallelograms between the same parallels and on the same base are equal in area. A diagonal of a parallelogram divides it into two triangles of the same area In this case area of (△ABC) = area of (△ADC). Also area of (△ABD) = area of (△BCD)
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Theorem 2
The area of a parallelogram is equivalent to the area of the rectangle of the same altitude and on the same base, i.e., between the same parallels. That is to say, the area of (||gm ABCD) = Area of (rectangle ABFE) since they lie between the same parallels AB and DE and on the same base.
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Theorem 3
The area of a triangle is half of the area of a parallelogram lying between the same parallels and on the same base. From the figure below, the area of (∆ APB) = ½ × Area of (||gm ABCD) seeing that they lie between the same parallels AB and PC and on the same base. Area of a parallelogram is the product of its base and the corresponding height.
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Theorem 4
Triangles between the same parallels as well on the same base are equivalent in area. Area of (∆ ABD) = Area of (∆ ABC) since they remain between the same parallels AB and DC and on the base AB. The area of a triangle is half the product of its corresponding height and any of its sides. This theorem is also called Heron's Theorem.
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Theorem 5
If a parallelogram and a parallelogram lies between the same parallels and on the same base, thus the area of the triangle will be equivalent to the half of the parallelogram.
Area of (△ABCD) = area of (△BCD)
Area of (||gmABCD) = AB × h
Area of (△ABE) = ½ AB × h
Area of (△ABE) = Area of (||gmABCD)
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Theorem 6
This is a Trapezium Area Axiom. According to this theorem, the area of a trapezium is half the product of the sum of its parallel sides and the altitude. A trapezium is a type of a quadrilateral that has two of its sides parallel to each other. There is also a type of trapezium which we call an isosceles trapezium whose non-parallel sides are equal. Having said that, suppose that we have ‘a’ and ‘b’ the parallel sides and ‘h’, the distance between the parallel sides of a parallelogram ABCD. Then Area = (½ A+B) × h
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Theorem 7
Triangles between the same parallels and on the same base share the same area.
Area of (△ABD) = ½ × AB × h
Area of (△ABC) = ½ × AB × h
Hence, Area of (△ABD) = Area of (△ABC)
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Theorem 8
Triangles with equal areas and whose one side of one of the triangles equivalent to one side of the other triangle, with their corresponding heights the same.
Theorem 9
Two triangles whose bases are the same (or equal bases) and equal area remain between the same parallels.
FAQs on Understanding Theorems on Area in Geometry
1. What are the theorems on area in geometry?
The theorems on area state that triangles on the same base and between the same parallels have equal area, and parallelograms on the same base and between the same parallels have equal area. These results are fundamental in coordinate geometry and Euclidean geometry.
- Triangle Theorem: Triangles on the same base and between the same parallels are equal in area.
- Parallelogram Theorem: Parallelograms on the same base and between the same parallels are equal in area.
2. What is the theorem of triangles on the same base and between the same parallels?
The theorem states that triangles on the same base and between the same parallels are equal in area. Since both triangles share the same base and have the same height (distance between the parallels), their areas are equal.
- Area of triangle = ½ × base × height
- Same base → equal base length
- Same parallels → equal height
3. What is the theorem of parallelograms on the same base and between the same parallels?
The theorem states that parallelograms on the same base and between the same parallels are equal in area. Because both share the same base and height, their areas are equal.
- Area of parallelogram = base × height
- Same base → equal base
- Same height (between parallels) → equal area
4. What is the formula for the area of a triangle used in area theorems?
The formula for the area of a triangle is Area = ½ × base × height. The height is the perpendicular distance from the base to the opposite vertex.
- Example: If base = 10 cm and height = 6 cm
- Area = ½ × 10 × 6 = 30 cm²
5. How do you prove that two triangles on the same base and between the same parallels are equal in area?
Two triangles on the same base and between the same parallels are equal in area because they have the same base and equal height. The proof follows directly from the area formula.
- Area = ½ × base × height
- Base is common
- Height (distance between parallels) is equal
6. Can you give an example of the area theorem with numbers?
Yes, if two triangles share a base of 8 cm and lie between the same parallels with height 5 cm, their areas are equal. Using the formula:
- Area = ½ × 8 × 5
- Area = 20 cm²
7. What is the difference between area theorem for triangles and parallelograms?
The difference is that triangles use the formula ½ × base × height, while parallelograms use base × height, but both follow the same principle of equal area between the same parallels.
- Triangle area depends on half the rectangle formed.
- Parallelogram area equals the full rectangle area.
8. Why do figures between the same parallels have equal height?
Figures between the same parallels have equal height because the perpendicular distance between parallel lines is constant. This fixed distance represents the height in area calculations.
- Parallel lines never meet.
- The shortest distance between them is always equal.
9. How are area theorems used in coordinate geometry?
Area theorems are used in coordinate geometry to compare triangle areas and prove collinearity of points. The coordinate formula for triangle area is:
- Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
10. What are common mistakes when applying the theorems on area?
A common mistake is confusing equal bases with equal heights or forgetting that the figures must lie between the same parallels. Key points to remember:
- Both base and height must be equal.
- Height must be perpendicular to the base.
- Figures must lie between the same pair of parallel lines.
