3.4-Some Special Parallelograms

3.4-Some Special Parallelograms Important Formulae

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3.4 - Some Special Parallelograms
  • Rectangle: A parallelogram with all angles equal to $90^\circ$.
  • Rhombus: A parallelogram where all sides are equal in length, but angles are not $90^\circ$.
  • Square: A parallelogram that is both a rectangle and a rhombus, with all sides equal and all angles $90^\circ$.
  • Properties: Opposite sides of a parallelogram are equal and parallel, and opposite angles are equal.
  • Area: Area of a parallelogram = Base × Height, where the base is any side and height is the perpendicular distance between opposite sides.

3.4 - Some Special Parallelograms

A parallelogram is a quadrilateral with opposite sides parallel and equal in length. Some special types of parallelograms have unique properties. In this section, we will explore three such special parallelograms: rhombus, rectangle, and square.

1. Rhombus

A rhombus is a type of parallelogram where all four sides are of equal length. The properties of a rhombus include:

  • Opposite sides are parallel.
  • Opposite angles are equal.
  • The diagonals bisect each other at right angles (90°).
  • The diagonals bisect the angles of the rhombus.
  • The area of a rhombus can be calculated using the formula: $ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $ where $d_1$ and $d_2$ are the lengths of the diagonals.
2. Rectangle

A rectangle is a parallelogram where all angles are 90°. The properties of a rectangle include:

  • Opposite sides are parallel and equal in length.
  • All angles are 90°.
  • The diagonals are equal in length and bisect each other.
  • The area of a rectangle is given by the formula: $ \text{Area} = l \times b $ where $l$ is the length and $b$ is the breadth of the rectangle.
3. Square

A square is a special type of parallelogram that is both a rhombus and a rectangle. In a square, all sides are equal, and all angles are 90°. The properties of a square include:

  • All four sides are equal in length.
  • All four angles are 90°.
  • The diagonals are equal in length, bisect each other at right angles, and are also perpendicular.
  • The area of a square can be calculated using the formula: $ \text{Area} = a^2 $ where $a$ is the length of a side of the square.
  • The perimeter of a square is given by: $ \text{Perimeter} = 4a $ where $a$ is the length of a side of the square.
4. Key Differences Between Rhombus, Rectangle, and Square

Below are the key differences between rhombus, rectangle, and square:

Property Rhombus Rectangle Square
All sides equal Yes No Yes
All angles 90° No Yes Yes
Diagonals equal No Yes Yes
Diagonals bisect each other at right angles Yes No Yes
5. Area Formulas

For each special parallelogram, the formula for area differs slightly:

  • Rhombus: $ \text{Area} = \frac{1}{2} \times d_1 \times d_2 $
  • Rectangle: $ \text{Area} = l \times b $
  • Square: $ \text{Area} = a^2 $

These formulas are important for calculating the area of the special types of parallelograms based on their respective properties.

State whether True or False.

(a) All rectangles are squares
(b) All rhombuses are parallelograms.
(c) All squares are rhombuses and also rectangles.
(d) All squares are not parallelograms.
(e) All kites are rhombuses.
(f) All rhombuses are kites.
(g) All parallelograms are trapeziums.
(h) All squares are trapeziums.

Solution:

(a) All rectangles are squares
False
(b) All rhombuses are parallelograms.
True
(c) All squares are rhombuses and also rectangles.
True
(d) All squares are not parallelograms.
False
(e) All kites are rhombuses.
False
(f) All rhombuses are kites.
False
(g) All parallelograms are trapeziums.
False
(h) All squares are trapeziums.
False

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Identify all the quadrilaterals that have.

(a) four sides of equal length.
(b) four right angles.

Solution:

(a) Four sides of equal length:

The quadrilaterals that have four sides of equal length are:

  • Square
  • Rhombus
(b) Four right angles:

The quadrilaterals that have four right angles are:

  • Rectangle
  • Square

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Explain how a square is.

(i) a quadrilateral
(ii) a parallelogram
(iii) a rhombus
(iv) a rectangle

Solution:

How a Square is:

(i) A quadrilateral: A square is a quadrilateral because it has four sides and four angles. The sum of its interior angles is $360^\circ$.

(ii) A parallelogram: A square is a parallelogram because its opposite sides are parallel and equal in length. Also, opposite angles are equal, and adjacent angles are supplementary (add up to $180^\circ$).

(iii) A rhombus: A square is a rhombus because all its sides are of equal length. Additionally, its diagonals bisect each other at right angles and are not of equal length (except in a square, they are equal).

(iv) A rectangle: A square is a rectangle because it has four right angles. Moreover, opposite sides are parallel and equal in length, which satisfies the definition of a rectangle.

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Name the quadrilaterals whose diagonals.

(i) bisect each other.
(ii) are perpendicular bisectors of each other.
(iii) are equal.

Solution:

Quadrilaterals whose diagonals:

(i) Bisect each other:

Answer: Parallelogram, Rectangle, Rhombus, Square

(ii) Are perpendicular bisectors of each other:

Answer: Rhombus, Square

(iii) Are equal:

Answer: Rectangle, Square

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Explain why a rectangle is a convex quadrilateral.

Solution:

Why is a Rectangle a Convex Quadrilateral?
A rectangle is a type of quadrilateral where all four angles are $90^\circ$. A convex quadrilateral is defined as a quadrilateral in which all of its interior angles are less than $180^\circ$ and the diagonals lie inside the quadrilateral. In the case of a rectangle: 1. All interior angles are $90^\circ$, which is less than $180^\circ$. 2. The diagonals of the rectangle intersect each other inside the shape. Therefore, a rectangle satisfies the condition of having all interior angles less than $180^\circ$, and its diagonals are inside the figure, making it a convex quadrilateral.

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ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

Solution:

Explanation of Why O is Equidistant from A, B, and C

In a right-angled triangle ABC, where $\angle ABC = 90^\circ$, the point O is the midpoint of the side opposite the right angle (i.e., side AC). We are required to show why point O is equidistant from points A, B, and C.

First, recall the property that the midpoint of the hypotenuse of a right-angled triangle is equidistant from all three vertices of the triangle. This property comes from the fact that in a right-angled triangle, the circumcenter (the center of the circumcircle) lies at the midpoint of the hypotenuse.

Now, consider the circumcircle of triangle ABC. The circumcenter of a right-angled triangle is always the midpoint of the hypotenuse. Since O is the midpoint of side AC, O must be the circumcenter of the triangle.

Since O is the circumcenter, it is equidistant from all three vertices of the triangle. Hence, the distances from O to A, O to B, and O to C are all equal.

This proves that O is equidistant from A, B, and C.

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