Areas of Parallelograms and Triangles • Topic 3 of 3

Geometric Relationships Between Shapes on the Same Base

What are the Area Relationships Between Shapes on the Same Base and Parallels?

Mathematicians discovered a fascinating rule when shapes share the exact same base line and sit nestled between the same pair of parallel lines.

Let us explore these key relationship rules:

  1. Parallelograms on the Same Base: If two or more parallelograms share the same base line and lie inside the same parallel lines, their surface areas will be perfectly equal. Why? Because they share the exact same base length, and the distance between the parallel lines ensures their vertical heights are completely identical!
  2. Triangles on the Same Base: Similarly, two triangles sharing the same base line and lying between the same parallel lines will have completely equal areas.
  3. The Triangle and Parallelogram Mix: If a triangle and a parallelogram sit on the same base line and lie between the same parallel lines, the area of the triangle will be exactly half of the area of the parallelogram.
Formula: Area(Triangle) = 0.5 × Area(Parallelogram)

These theorems are incredibly helpful because they allow you to instantly determine the area of a highly distorted, slanted shape simply by comparing it to a simpler straight shape next to it!

Area Theorems — Key ResultsMedian divides △ into 2 equal area trianglesIf AD is median of △ABC, ar(△ABD) = ar(△ACD)1Diagonal of parallelogram bisects areaDiagonal AC divides ▱ABCD into 2 equal △s2Same base, same parallels → equal areaar(△ABP) = ar(△ABQ) if P, Q on same parallel3Converse: Equal area + same base → same parallelsIf equal area on same base → vertices are collinear4Remember: Area of rhombus = ½ × d₁ × d₂ (product of diagonals)Parallelogram and rectangle on same base and height have equal area
Solved Examples
1
Worked Example

Solve a standard problem on Geometric Relationships Between Shapes on the Same Base.

Solution

Apply the formula/method shown in the concept section above.

Key Points

  • Understand the definition and properties of Geometric Relationships Between Shapes on the Same Base.
  • Study the worked examples and practice similar problems.
  • Always verify your answer using the original conditions.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Parallelograms on the same base and between the same parallels are equal in:
Explanation: Equal area.
Q2.Triangles on the same base and between the same parallels are equal in:
Explanation: Equal area.
Q3."Between the same parallels" means the figures have the same:
Explanation: Same height.
Q4.A median of a triangle divides it into two triangles of:
Explanation: Equal area.
Practice more MCQs → 📝 Assignment · 35 marks