Class 9 Mathematics Notes Chapter 9 (Areas of Parallelograms and Triangles) – Mathematics Book

Alright class, let's focus on Chapter 9: Areas of Parallelograms and Triangles. This is a crucial chapter, not just for your Class 9 understanding, but also because the concepts frequently appear in geometry sections of various government exams. We need to understand how areas relate, especially when figures share a common base and lie between the same parallel lines.

Chapter 9: Areas of Parallelograms and Triangles - Detailed Notes

1. Introduction: Area Concept

• Planar Region: The part of the plane enclosed by a simple closed figure is called a planar region.
• Area: The measure (magnitude) of the planar region enclosed by a closed figure is called its area. It's measured in square units (like cm², m²).
• Congruence and Area:
  • If two figures are congruent, they must have equal areas.
  • However, the converse is not necessarily true: two figures with equal areas need not be congruent. (Example: A rectangle of 6x2 and a square of side √12 both have area 12 sq units, but are not congruent).

2. Figures on the Same Base and Between the Same Parallels

• Definition: Two figures are said to be on the same base and between the same parallels if:

  • They have a common side (the base).
  • The vertices (or the vertex) opposite to the common base lie on a line parallel to the base.

• Identification: This is key. Look for a shared line segment as the base and check if the highest points of both figures lie on a single line parallel to that base.

  (Imagine two buildings standing on the same road segment, and their rooftops lie along a straight line parallel to the road - they are on the same base and between the same parallels.)

3. Parallelograms on the Same Base and Between the Same Parallels

• Theorem 9.1: Parallelograms on the same base and between the same parallels are equal in area.
  • Explanation: If parallelogram ABCD and parallelogram EBCF share base BC and lie between parallels BC and AF, then Area(ABCD) = Area(EBCF).
  • Significance: This allows us to equate areas without calculating them directly, just by observing their geometric configuration.
  • Corollary (Area of a Parallelogram): Area of a parallelogram = Base × Corresponding Height (Altitude).
    • The theorem helps establish this. Since a rectangle is also a parallelogram, a parallelogram on base 'b' and between parallels 'h' distance apart will have the same area as a rectangle with sides 'b' and 'h'. Area = b × h.

4. Triangles on the Same Base and Between the Same Parallels

• Theorem 9.2: Triangles on the same base (or equal bases) and between the same parallels are equal in area.

  • Explanation: If ΔABC and ΔDBC share base BC and lie between parallels BC and AD, then Area(ΔABC) = Area(ΔDBC).
  • Significance: Similar to parallelograms, this relates triangle areas based on their position.

• Theorem 9.3 (Converse of 9.2): Triangles on the same base (or equal bases) and having equal areas lie between the same parallels.

  • Explanation: If Area(ΔABC) = Area(ΔDBC) and they share base BC, then the line AD must be parallel to BC.

5. Relationship Between Parallelogram and Triangle Areas (Same Base & Parallels)

• Key Result: If a parallelogram and a triangle are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram.
  • Explanation: Consider parallelogram ABCD and triangle EBC on base BC and between parallels BC and AE.
    Area(ΔEBC) = ½ × Base × Height = ½ × BC × h
    Area(parallelogram ABCD) = Base × Height = BC × h
    Therefore, Area(ΔEBC) = ½ × Area(parallelogram ABCD).
  • Corollary (Area of a Triangle): Area of a triangle = ½ × Base × Corresponding Height (Altitude). This formula is directly linked to the parallelogram area formula via this relationship.

6. Median of a Triangle and Area

• Definition: A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
• Property: A median of a triangle divides it into two triangles of equal areas.
  • Explanation: If AD is the median of ΔABC (D is midpoint of BC), then Area(ΔABD) = Area(ΔACD).
  • Reason: Both triangles (ABD and ACD) have equal bases (BD = DC) and the same height (the perpendicular distance from vertex A to the base BC). Area = ½ × base × height.

Key Takeaways for Exams:

• Be able to quickly identify figures on the same base and between the same parallels.
• Memorize and understand Theorems 9.1 and 9.2 – they are fundamental for comparing areas.
• Remember the relationship: Area(Triangle) = ½ Area(Parallelogram) when they are on the same base and between the same parallels.
• Know the area formulas: Parallelogram (b × h) and Triangle (½ × b × h).
• Understand the property of the median dividing a triangle into two triangles of equal areas. This is often used in multi-step problems.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on this chapter, designed for practice:

1. The area of a figure is a measure of the ______ enclosed by it.
(A) Perimeter
(B) Planar region
(C) Diagonal length
(D) Volume

2. Two parallelograms PQRS and PQMN stand on the same base PQ and between the same parallels PQ and RN. Which of the following is true?
(A) Area(PQRS) = 2 × Area(PQMN)
(B) Area(PQRS) = Area(PQMN)
(C) Area(PQMN) = 2 × Area(PQRS)
(D) Area(PQRS) = ½ × Area(PQMN)

3. If ΔABC and ΔDBC are on the same base BC and between the same parallels AD and BC, then:
(A) Area(ΔABC) > Area(ΔDBC)
(B) Area(ΔABC) < Area(ΔDBC)
(C) Area(ΔABC) = Area(ΔDBC)
(D) Area(ΔABC) = ½ Area(ΔDBC)

4. A parallelogram ABCD and a triangle ABE share the same base AB and lie between the same parallels AB and DE. If the area of the triangle ABE is 25 cm², what is the area of the parallelogram ABCD?
(A) 25 cm²
(B) 12.5 cm²
(C) 50 cm²
(D) 625 cm²

5. The area of a parallelogram is 72 cm² and its altitude is 8 cm. What is the length of its corresponding base?
(A) 18 cm
(B) 6 cm
(C) 9 cm
(D) 12 cm

6. A triangle has a base of 10 cm and a corresponding height of 6 cm. Its area is:
(A) 60 cm²
(B) 30 cm²
(C) 15 cm²
(D) 16 cm²

7. AD is a median of ΔABC. If the area of ΔABC is 40 cm², what is the area of ΔABD?
(A) 40 cm²
(B) 10 cm²
(C) 80 cm²
(D) 20 cm²

8. Two triangles, ΔPQR and ΔSQR, have the same base QR and equal areas. What can be concluded about the line segment PS?
(A) PS is perpendicular to QR
(B) PS is parallel to QR
(C) PS bisects QR
(D) PS = QR

9. ABCD is a parallelogram. P is any point on CD. Area(ΔAPB) is related to Area(parallelogram ABCD) as:
(A) Area(ΔAPB) = Area(ABCD)
(B) Area(ΔAPB) = ½ Area(ABCD)
(C) Area(ΔAPB) = 2 × Area(ABCD)
(D) Area(ΔAPB) = ¼ Area(ABCD)
(Hint: Think about ΔAPB and parallelogram ABCD relative to base AB)

10. In parallelogram ABCD, diagonals AC and BD intersect at O. Which statement is always true?
(A) Area(ΔAOB) = Area(ΔBOC) = Area(ΔCOD) = Area(ΔDOA)
(B) Area(ΔABC) = Area(ΔADC)
(C) Both (A) and (B) are true
(D) Only (B) is true

Answer Key for MCQs:

1. (B)
2. (B)
3. (C)
4. (C) - Area(Parallelogram) = 2 * Area(Triangle)
5. (C) - Area = base * height => 72 = base * 8 => base = 9
6. (B) - Area = ½ * base * height = ½ * 10 * 6 = 30
7. (D) - Median divides triangle into two equal areas. 40 / 2 = 20
8. (B) - Theorem 9.3 (Converse)
9. (B) - ΔAPB and parallelogram ABCD are on the same base AB and between same parallels AB and CD.
10. (C) - Diagonals divide a parallelogram into 4 triangles of equal area (A), and a diagonal divides it into two congruent (hence equal area) triangles (B).

Study these notes carefully, understand the theorems, and practice identifying the conditions (same base, same parallels). Good luck with your preparation!