Complete notes with theorems, proofs, and formula sheet
This chapter focuses on calculating areas of parallelograms and triangles, particularly when they share the same base and lie between the same parallel lines. These concepts are fundamental for understanding geometric relationships.
Area: The measure of the surface enclosed by a closed figure.
Congruent Figures: Have equal areas (and same shape/size).
Equal Figures: Have equal areas (but may have different shapes).
Note: All congruent figures are equal, but not all equal figures are congruent.
Two figures are on the same base and between the same parallels if:
• They have a common base (side)
• The vertices opposite to the base lie on a line parallel to the base
Area = Base × Height
Where height is the perpendicular distance between the base and its opposite side.
Parallelograms on the same base and between the same parallels are equal in area.
Area of parallelogram = Base × Height (perpendicular distance between parallels)
If a parallelogram and a rectangle are on the same base and between the same parallels, they have equal areas.
Q: A parallelogram has a base of 12 cm and height of 8 cm. Find its area.
Solution:
Area = Base × Height
= 12 × 8
= 96 cm²
Q: Two parallelograms ABCD and ABEF are on the same base AB and between same parallels AB and CF. If area of ABCD = 60 cm², what is area of ABEF?
Solution:
By Theorem 1, parallelograms on same base and between same parallels have equal areas.
Area of ABEF = 60 cm²
Area = ½ × Base × Height
Where height is the perpendicular distance from the base to the opposite vertex.
Triangles on the same base and between the same parallels are equal in area.
The area of a triangle is half the area of a parallelogram on the same base and between the same parallels.
Area of △ = ½ × Area of parallelogram (when on same base and between same parallels)
Q: A triangle has base 10 cm and height 6 cm. Find its area.
Solution:
Area = ½ × Base × Height
= ½ × 10 × 6
= ½ × 60
= 30 cm²
Q: A parallelogram ABCD and triangle ABE have the same base AB and are between same parallels. If area of parallelogram is 80 cm², find area of triangle.
Solution:
By Theorem 4:
Area of triangle = ½ × Area of parallelogram
= ½ × 80
= 40 cm²
A median of a triangle divides it into two triangles of equal area.
Each smaller triangle has area = ½ × area of original triangle.
Q: In triangle ABC, D is the midpoint of BC. If area of ABC is 48 cm², find area of triangle ABD.
Solution:
AD is a median (connects vertex A to midpoint D of BC)
By median theorem, AD divides triangle into two equal areas
Area of △ABD = ½ × Area of △ABC
= ½ × 48
= 24 cm²
| Concept | Result |
|---|---|
| Parallelograms (same base, same parallels) | Equal areas |
| Triangles (same base, same parallels) | Equal areas |
| Triangle vs Parallelogram (same base, same parallels) | Triangle area = ½ × Parallelogram area |
| Median of triangle | Divides into two equal areas |
| Diagonal of parallelogram | Divides into two equal triangles |
Area = Base × Height
A = b × h
Height is perpendicular distance
Area = ½ × Base × Height
A = ½ × b × h
Half of parallelogram formula
Parallelograms → Equal areas
Triangles → Equal areas
Triangle = ½ × Parallelogram
Fundamental theorem
Median divides triangle
Two equal areas created
Each = ½ × original area
Important for problems
Diagonal of parallelogram
Creates 2 congruent triangles
Each = ½ × parallelogram area
Equal areas
Area = Length × Width
A = l × w
Special parallelogram
Area = Side × Side
A = s²
All sides equal
• Always find perpendicular height
• Same base & parallels → equal areas
• Triangle = ½ parallelogram
Problem-solving strategy
• Always draw clear diagrams showing base and height.
• Mark parallel lines clearly on your diagrams.
• Remember: height is ALWAYS perpendicular to the base.
• "Same base and between same parallels" is a key phrase - memorize what it means!
• Triangle area is always half of parallelogram area (same base, same parallels).
• Don't confuse slant height with perpendicular height.
• Practice identifying which figures share the same base and parallels.
• Median problems are common - remember it divides area equally!
Q1: Parallelogram ABCD has base 15 cm and area 120 cm². Find its height.
Solution: Area = base × height → 120 = 15 × h → h = 8 cm
Q2: A triangle and parallelogram have the same base 10 cm and are between same parallels. If triangle area is 35 cm², find parallelogram area.
Solution: Triangle area = ½ × Parallelogram area
35 = ½ × P → P = 70 cm²
Q3: Triangle PQR has area 60 cm². S is midpoint of QR. Find area of triangle PQS.
Solution: PS is median → divides into equal areas
Area of PQS = ½ × 60 = 30 cm²