NCERT Class 9 Maths · Chapter 9
NCERT Solutions Class 9 Maths Chapter 9 — Areas of Parallelograms & Triangles
Step-by-step solutions for all exercises in NCERT Class 9 Maths Areas of Parallelograms & Triangles.
Chapter Overview
Prove theorems relating areas of figures on the same base and between the same parallels.
This chapter is part of the NCERT Mathematics textbook for Class 9 and is important for CBSE school examinations. The concepts covered here build the foundation for more advanced topics in higher classes.
Below you will find solved examples from this chapter. Each solution includes detailed step-by-step working so you can understand the method, not just the answer.
Solved Examples from Areas of Parallelograms & Triangles
1For two planar figures to be considered 'between the same parallels', what is the essential condition they must satisfy?
Answer: A
Solution:
Step 1: The definition states that if two figures have a common base (or equal bases) and the vertices opposite to the base of each figure lie on a line parallel to the base, then they are said to be on the same base and between the same parallels.
Step 2: Option A directly captures this definition, explaining the geometric arrangement required.
2In a figure, parallelogram ABCD and triangle EBC are drawn such that point E lies on the side AD. For these two figures to be on the same base and between the same parallels, which condition is necessary?
Answer: B
Solution:
Step 1: The common base for both parallelogram ABCD and triangle EBC is BC.
Step 2: For them to be between the same parallels, the line containing the vertex opposite to the base (AD for the parallelogram, and E for the triangle, which lies on AD) must be parallel to the base BC.
Step 3: Therefore, the line AD must be parallel to BC.
3Parallelogram PQRS and parallelogram MNRS are on the same base SR and between the same parallels SR and PM. If Area(PQRS) = 45 cm², what is Area(MNRS)?
Answer: B
Solution:
Step 1: According to Theorem 9.1, parallelograms on the same base and between the same parallels are equal in area.
Step 2: Given that PQRS and MNRS are on the same base SR and between the same parallels SR and PM, their areas must be equal.
Step 3: Therefore, Area(MNRS) = Area(PQRS).
Step 4: Area(MNRS) = 45 cm².
4Which of the following conditions is *not necessarily true* for two triangles on the same base and between the same parallels?
Answer: C
Solution:
Step 1: If two triangles are on the same base and between the same parallels, it implies they have the same base length and the same height corresponding to that base.
Step 2: Based on the formula Area = 1/2 × base × height, their areas will be equal (Option B is true). Their heights are indeed equal (Option A is true). And by definition, their opposite vertices lie on the same parallel line (Option D is true).
Step 3: However, having the same base and height does not guarantee that the other two sides of the triangles are equal. Therefore, their perimeters are not necessarily equal.
5A triangle and a parallelogram are on the same base and between the same parallels. If the area of the parallelogram is 60 cm², what is the area of the triangle?
Answer: A
Solution:
Step 1: According to a key theorem, if a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram.
Step 2: Given Area(parallelogram) = 60 cm².
Step 3: Area(triangle) = 1/2 × Area(parallelogram).
Step 4: Area(triangle) = 1/2 × 60 cm² = 30 cm².
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