Page 228 - ICSE Math 8

Each diagonal of a parallelogram divides it into two congruent triangles. Thus, the area of D C
a parallelogram is equal to the sum of the areas of two triangles. In the adjoining figure,
diagonal AC divides parallelogram ABCD into two congruent triangles ABC and ADC.
∴ Area of parallelogram ABCD = Area of ∆ABC + Area of ∆ADC
= 2(Area of ∆ABC) or = 2(Area of ∆ADC) A B
D 24 cm C
Example 7: The adjacent sides of a parallelogram are 24 cm and 26 cm.
If the distance between the shorter sides is 18 cm, find its area. 18 cm 26 cm
Solution: Area of parallelogram ABCD = AB × DE = 24 cm × 18 cm = 432 cm 2
A E B

Example 8: In a parallelogram ABCD, AB = 12 cm, BC = 10 cm and CA = 18 cm. Find the area of the
parallelogram.
Solution: Area of parallelogram ABCD = 2 × Area of ∆ABC D C
In ∆ABC,
12 + 10 + 18 18 cm 10 cm
s = 2 cm = 20 cm

Area of ∆ABC = 20(20 12)(20 10)(20 18) cm− − − 2 A 12 cm B

7
2
2
= 20 8 10 2 cm× × × 2 = 2 ×× 3 2 5 2 cm×× 2 = 2 × 5 cm 2
5 2 ×
2
3
= 2 × 5 2 cm = 40 2 cm 2
2
∴ Area of parallelogram ABCD = 2 × 40 2 cm = 80 2 cm 2
Area of a Rhombus A B Maths Info
1
Area of rhombus ABCD = × Product of its diagonals A rhombus is a parallelogram
2
1 O a whose all sides are equal and
= 2 × AC × BD diagonals bisect each other at
Also, area of rhombus ABCD D C right angles.
= (Area of ∆AOB) + (Area of ∆BOC) + (Area of ∆COD) + (Area of ∆DOA)

Example 9: In a rhombus ABCD, AC = 8 cm and BD = 6 cm. Find its area and perimeter.
1 A D
Solution: Area of rhombus ABCD = × AC × BD 4 cm 3 cm
2
1
= 2 × 8 cm × 6 cm = 24 cm 2 3 cm 4 cm

In right-angled ∆AOB, O
1 1 1 1 B C
OA = AC = × 8 cm = 4 cm, and OB = BD = × 6 cm = 3 cm
2
2
2
2
(  Diagonals of a rhombus bisect each other)
2
2
AB = OA + OB ⇒ 2 AB = 4 + 2 3 cm = 16 9 cm+ = 25 cm = 5 cm
∴ Perimeter of rhombus ABCD = 4 × side = 4 × AB = 4 × 5 cm = 20 cm
D C
Example 10: Each side of a rhombus is of length 50 cm and the length of one of its
diagonals is 80 cm. Find the length of the other diagonal and the area of 40 cm
the rhombus. 40 cm O
Solution: Diagonals of a rhombus bisect each other at right angles. Therefore,

AO = OC = 40 cm and BO = OD. A 50 cm B

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