Solved Examples on Parallelogram
We have solved some select questions on parallelogram down below. They will help you better improve your understanding of the concepts in this chapter.
Example 1: Find the length of the other side of a parallelogram with a base of 12 cm and a perimeter of 60 cm.
Solution:
Given perimeter of a parallelogram = 60cm.
Base length of given parallelogram = 12 cm.
P = 2 (a + b) units
Where b = 12cm and P = 40cm.
60 = 2 (a + 12)
60 = 2a + 24
2a = 60-24
2a = 36
a = 18cm
Therefore, the length of the other side of the parallelogram is 18 cm.
Example 2: Find the perimeter of a parallelogram with the base and side lengths of 15cm and 5cm, respectively.
Solution:
Base length of given parallelogram = 15 cm
Side length of given parallelogram = 5 cm
Perimeter of a parallelogram is given by,
P = 2(a + b) units.
Putting the values, we get
P = 2(15 + 5)
P = 2(20)
P = 40 cm
Therefore, the perimeter of a parallelogram will be 40 cm.
Example 3: The angle between two sides of a parallelogram is 90°. If the lengths of two parallel sides are 5 cm and 4 cm, respectively, find the area.
Solution:
If one angle of the parallelogram is 90°. Then, the rest of the angles are also 90°. Therefore, the parallelogram becomes a rectangle. The area of the rectangle is length times breadth.
Area of parallelogram = 5 × 4
Area of parallelogram = 20cm2
Example 4: Find the area of a parallelogram when the diagonals are given as 8 cm, and 10 cm, the angle between the diagonals is 60°.
Solution:
In order to find the area of the parallelogram, the base and height should be known, lets’s first find the base of the parallelogram, applying the law of cosines,
b2 = 42 + 52 – 2(5)(4)cos(120°)
b2 = 16 + 25 – 40(0.8)
b2 = 9
b = 3cm
Finding the height of the parallelogram,
4/sinθ = b/sin120
4/sinθ = 3/-0.58
sinθ = -0.773
θ = 50°
Now, to find the height,
Sinθ = h/10
0.76 = h/10
h = 7.6cm
Area of the parallelogram = 1/2 × 3 × 7.6
= 11.4 cm2
Example 5: Prove that a parallelogram circumscribing a circle is a rhombus.
Solution:
Given:
- Parallelogram ABCD
- Circle PQRS
To prove: ABCD is a rhombus.
Proof:
We know that the tangents drawn from an exterior point to a circle are equal to each other. Therefore:
AP = AS ⇢ (1)
BP = BQ ⇢ (2)
DS = DR ⇢ (3)
CR = CQ ⇢ (4)
Adding the LHS and RHS of equations 1, 2, 3, and 4:
AP + BP + DS + CR = AS + BQ + DR + CQ
AB + DR + CR = AS + DS + BC
AB + CD = AD + BC
Since the opposite angles of a parallelogram are equal:
2AB = 2BC
AB = BC, and similarly, CD = AD.
Therefore: AB = CD = BC = AD.
Since all the sides are equal, ABCD is a rhombus.
Introduction to Parallelogram: Properties, Types, and Theorem
Parallelogram is a two-dimensional geometrical shape whose opposite sides are equal in length and parallel. The opposite angles of a parallelogram are equal in measure.
In this article, we will learn about the definition of a parallelogram, its properties, types, theorem and formulas on the area and perimeter of a parallelogram in detail.
Table of Content
- Parallelogram Definition
- Shape of Parellelogram
- Angles of Parallelogram
- Properties of Parallelogram
- Types of Parallelogram
- Parallelogram Formulas
- Area of Parallelogram
- Perimeter of Parallelogram
- Parallelogram Theorem
- Difference Between Parallelogram and Rectangle
- Solved Examples on Parallelogram
- Real-Life Examples of a Parallelogram
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