Exploring Rectangle and its Theorems – Definition, solved examples and  application

A rectangle is a four-sided polygon with all four angles measuring 90 degrees. It is also a type of parallelogram, which means that opposite sides are parallel and congruent. In short, if the two diagonals of a parallelogram bisect each other and congruent then, it is a rectangle.rectangle and its theorems

The properties of a rectangle can be used to prove several theorems. Here are some important theorems related to rectangles:

Some of the key theorems associated with rectangles include:

• The opposite sides of a rectangle are congruent.
• The rectangle has congruent diagonals.
• The diagonals of a rectangle bisect each other (application of the midpoint theorem).
• The perimeter of a rectangle is two times the sum of its length and width.
• The area of a rectangle is equal to the product of its length and width.
• Each angle of rectangles is 90°(application of Pythagorean Theorem)
• The sum of all interior angles of rectangles is 360°
• A diagonal of a rectangle is the diameter of its circumcircle (means circle passes through all the vertices of the triangle)
• The length of the diagonal is given by sqrt(l² +w²) ( application of Pythagorean theorem)
• The diagonals of the rectangle bisect each other at two different angles.(midpoint theorem) .One angle is acute and another angle beside it is obtuse.

Rectangle has congruent diagonals

This theorem states that the two diagonals of a rectangle are equal in length. It can be proven using the Pythagorean Theorem. Let's assume that the length of the rectangle is 'a' and the width is 'b'. The diagonal represented by the hypotenuse of a right triangle with legs 'a' and 'b'. Using the Pythagorean theorem, we can find that the length of the diagonal is √(a² + b²)

Example 1: The length of a rectangle is 10 cm and the width is 6 cm. Prove that the opposite sides are congruent.

Solution: Let's label the rectangle as ABCD, where AB is the length and BC is the width. We need to prove that AB is congruent to CD.

We know that opposite sides of a rectangle are parallel, so AB is parallel to CD. Let's draw a diagonal of the rectangle, AC, which divides the rectangle into two congruent right triangles, ABC and ACD.

Since triangle ABC is a right triangle, we can use the Pythagorean theorem to find the length of AC:

AC²= AB²+ BC²

= 10²+ 6²

= 100 + 36

= 136

AC = sqrt(136) = 11.66 cm (rounded to two decimal places)

Since triangle ACD is congruent to triangle ABC, we have:

BD = AC = 11.66 cm and AB = CD

Therefore, opposite sides of the rectangle are congruent, and we have proven that AB = CD.

Example 2:  In the given rectangle ABCD, find x also find the unknown sides.

Solution : As we know, opposite sides of rectangles are congruent.

AB = CD

4x – 6 = x+ 3

4x – x = 3+6

3x = 9

x = 3

AB= 4x -6

Plug in the value of x=3 in above equation

AB = 4(3) – 6 = 6 cm

CD = x+3

Plug in x= 3 in the above equation

CD = 3 +3  = 6cm

Answer: AB = CD = 6

This proves that opposite sides of rectangles are equal and congruent.

Example 3: In rectangle ABCD, length is 4 cm and width is 3 cm, then prove that the diagonals of rectangles are congruent.

Solution: As we know that opposite sides of rectangles are congruent, so

AB= CD = 4cm and AD= BC= 3 cm.

Also, we know that each angle of rectangle are right angle, hence, by Pythagorean theorem

AC = sqrt( AB²+ BC²)

AC = sqrt ( 4² + 3²)

AC = sqrt(25)

AC = 5 cm.

Similarly, if we consider triangle ADC, we will get BD = 5cm.

This proves that the diagonals of the rectangle are congruent.

The perimeter of a rectangle can be expressed as 2(l + w) :

This theorem states that the perimeter of a rectangle can be calculated by adding two times  the length and twice the width.

Example 4: If a rectangle PQRS has a length of 10 cm and a width of 4 cm.

Perimeter = 2(l + w)

P = 2(10 + 4)

= 2 x 14

P = 28 cm

Example 5: In rectangle ABCD, AB = 8 cm and CD = 8 cm. Find the perimeter of the rectangle.

Solution: Since AB and CD are opposite sides of the rectangle, they are congruent. Therefore, the perimeter of the rectangle is:

P = (AB + CD + BC + AD)

= (8 + 8 + BC + AD)

= (16 + BC + AD)

We need to find the values of BC and AD to calculate the perimeter. Since opposite sides of a rectangle are parallel, we know that BC is congruent to AD. Let's assume that BC = x cm. Then, we have:

AD = x cm (by opposite sides are congruent)

AB = CD = 8 cm (given)

Since the sum of the interior angles of a rectangle is 360 degrees, we have:

angle A + angle B + angle C + angle D = 360

Since all four angles of a rectangle are right angles (90 degrees), we have:

90 + 90 + 90 + 90 = 360

Therefore, the perimeter of the rectangle is:

P = (16 + BC + AD)

= (16 + x + x)

= (16 + 2x)

= 16 + 2x

Answer : P = 16 + 2x

Application: The Perimeter Theorem is used in architecture to calculate the amount of material needed to enclose a rectangular building or room.

The Area Theorem: rectangle and its theorems

The area of a rectangle is equal to the product of its length and width.

Example 6: If a rectangle has a length of 6 units and a width of 8 units, then

find the area of rectangle ABCD

Solution : L= 6 units and w = 8 units

A = lw

A = 6 x 8 = 48 square units.

Application: The Area Theorem is used in geometry to find the area of shapes such as parallelograms and trapezoids, which can be divided into rectangles.

The Midpoint Theorem:

The midpoint of a rectangle is equidistant from its vertices. The diagonal bisecting each other can be proved by the midpoint theorem.

Example 7: If a rectangle ABCD has vertices at  A(0,0), B(6,0), C(6,8), and D(0,8), then find the coordinates of the midpoint and prove that the midpoint is equidistant from its vertices.

Solution: Join the diagonal AC and let the co-ordinates of mid point of AC be O(x,y)

So according to the mid point formula x= (x1+ x2)/ 2 and

y = (y1+ y2)/2

X= (0+6)/2 = 3 and y =(0+8)/2 = 4

So the co-ordinates of O(3,4)

Similarly, if we consider the diagonal BD, then we will get the coordinates of O again (3,4). This proves that point O is equidistant from all 4 vertices of the rectangle. This proves that DO = OB and AO = OC diagonals bisect each other

Example 8 :  In rectangle ABCD, AO = 5x-2 and OC= 10, find x, also find the length of the diagonals.

Solution: As we know that the diagonals of the rectangle bisect each other so,

AO = OC

5x -2 = 3x+10

Bring the like terms together using the opposite operation

5x – 3x = 10 +2

2x = 12

Now to make x alone, divide both sides by 2

x= 6

Now plug in the value of x in AO and OC

AO = 5x -2 = 5(6) – 2 = 30 – 2 = 28 cm

OC = 3x +10 = 3(6) + 10 = 28 cm

So the diagonal AC = AO + OC

AC = 28 + 28 = 56

As the diagonals of rectangles are congruent so

AC = BD

So BD = 56 cm

Application: The Midpoint Theorem is used in navigation to find the midpoint between two points on a map or GPS.

The Pythagorean Theorem:

The Pythagorean theorem can be used to find the length of the diagonal of a rectangle.

Example 9: If a rectangle ABCD has a length of 12 units and a width of 9 units, then find the  length of the diagonal AC .

Solution: As ABCD is a rectangle, so all angles are right angles. In triangle ABC, B = 900

By Pythagorean theorem,

AC = sqrt(l² + w²)

= sqrt(12² + 9²)

AC= 15 units.

Example 10 : Find the length of  AD in rectangle ABCD. AB= 16 cm and  BD= 20 cm

Solution:  As we know that triangle ABD is a right triangle so according to the Pythagorean theorem,

BD² = AB² + AD²

20² = 16² + AD²

400 = 256 + AD²

AD² = 400 -256

AD = sqrt(144)

AD = 12 cm.

Application: The Pythagorean Theorem is used in engineering to calculate the force required to bend or break rectangular objects. It is also used in navigation to find the distance between two points on a map or GPS.

In summary, a rectangle is a four-sided polygon with all four angles measuring 90 degrees. Its properties can be used to prove several theorems, including the congruence of diagonals, the parallelism and congruence of opposite sides, the congruence of opposite angles, and formulas for the perimeter and area of a rectangle.

Rectangles and their theorems have significant importance in mathematics and our daily lives. Here are some reasons why:

Logical reasoning: Proving theorems related to rectangles requires logical reasoning and critical thinking skills. These skills are important in all areas of life, not just in mathematics.

Problem-solving: Theorems related to rectangles are often used in problem-solving situations, both in math and in real-life scenarios. For example, knowing that opposite sides of a rectangle are congruent can help in calculating the perimeter of a rectangle when only some of the sides are known.

Practical applications: Rectangles are commonly found in everyday objects such as books, picture frames, and computer screens. The properties of rectangles and their theorems are essential for designing and constructing these objects.

Geometry: Rectangles are used in geometry to study the properties of 2D shapes. They are the most basic quadrilateral shape, and many other shapes can be formed from them. Theorems related to rectangles help to understand concepts like area, perimeter, diagonal length, and angles of polygons.

Construction: Rectangles have a very simple construction, which makes them useful in many fields. For example, they are used in construction to create square corners in buildings, frames, and windows.

Engineering: Engineers use rectangles and their properties to design many types of structures, such as bridges, dams, and buildings. The theorems related to rectangles help engineers to calculate the dimensions and forces acting on these structures.

Design: Rectangles are useful in graphic design, web design, and other visual arts. They provide a stable and consistent foundation for layout and design and are easily combined with different shapes to create complex designs.

Mathematics education: Rectangles and their theorems are useful in teaching students about basic geometry and algebraic concepts. They are used as a stepping stone to more complex concepts like trigonometry, calculus, and analytic geometry

In conclusion, rectangle and its theorems are fundamental concepts in mathematics and have many practical applications in different fields. Understanding their properties and theorems related to them is essential in many areas of education and professional life.