Mathematics

Parallelogram Properties and Area

93 Questions

Parallelogram properties and area questions test the ability to calculate dimensions using base and height ratios. Problems also cover diagonal properties and internal angles. These geometry concepts regularly appear in quantitative aptitude sections of various exams.

Area calculationBase and height ratiosDiagonal propertiesInterior anglesGeometric theorems

Parallelogram Properties and Area Questions

Multiple choice maths constructions mid-point formula midpoints division of a line segment

The end points of a diagonal of a parallelogram are $(1, 3)$ and $(5, 7)$, then the mid-point of the other diagonal is ..........

  1. $(1, 7)$
  2. $(3, 5)$
  3. $(5, 3)$
  4. $(7, 1)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given end points of diagonal of parallelogram $(1,3)$ and $(5,7)$

We know that the mid point of two point $(x _1,y _1) \ and \  (x _2,y _2)$ are
$\dfrac{x _{1}+x _{2}}{2},\dfrac{y _{1}+y _{2}}{2}$
$\Rightarrow \dfrac{1+5}{2},\dfrac{3+7}{2}$
$\Rightarrow( 3,5)$

Multiple choice maths constructions mid-point formula midpoints division of a line segment

If two vertices of a parellelogram are $(3,2)$ and $(-1,0)$ and the diagonals intersect at $(2, -5)$, then the other two vertices are:

  1. $(1, -10),(5, -12)$
  2. $(1, -12),(5, -10)$
  3. $(2, -10),(5, -12)$
  4. $(1, -10),(2, -12)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the given vertices be $ A (1,-12)  B (5, -10)  C(a,b) ; D(x,y) $

We know that the diagonals of a parallelogram bisect each other. So,the midpoint of AC $ = (2,-5) $ and of BD $ = (2,-5) $. 

Mid point of two points $ { (x } _{ 1 },{ y } _{
1 }) $ and $ { (x } _{ 2 },{ y } _{ 2 }) $ is  calculated by the formula

$ \left( \frac { { x } _{ 1 }+{ x } _{ 2 } }{ 2 } ,\frac { { y } _{ 1 }+y _{ 2 }

}{ 2 }  \right) $
Hence, midpoint of $ AC = (2,-5) $
$ => \left( \frac {1 + a }{ 2 } ,\frac { -12 + b }{ 2 }  \right) \quad = (2,-5) $
$ => \frac {1 + a }{ 2 } = 2 ; \frac { -12 + b }{ 2 }

= -5 $
$ => 1+ a = 4 ; -12 + b = -10 $
$ a = 3 ; b = 2 $

Hence, C $ =(3,2) $

And, midpoint of $ BD = (2,-5) $
$ => \left( \frac {5 + x}{ 2 } ,\frac { -10 + y }{ 2 }  \right) \quad = (2,-5) $
$ => \frac {5 + x }{ 2 } = 2 ; \frac { -10 + y }{ 2 }

= -5 $
$ => 5 + x = 4 ; -10 + y = -10 $
$ x = -1 ; y = 0 $

$ => D  =(-1,0) $


Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

The ratio of the measures of the consecutive angles of a quadrilateral is 1:2:3:4. What type of quadrilateral is it

  1. Trapezium

  2. Kite

  3. Parallelogram

  4. Rectangle
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of angles in a quadrilateral is 360 degrees. If the ratio is 1:2:3:4, the angles are 36, 72, 108, and 144 degrees. Since one pair of consecutive angles (72+108) sums to 180, it is a trapezium.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

If the angles A, B, C and D of a quadrilateral ABCD in the same order are in the ratio 3 : 7 : 6 : 4, then ABCD is a 

  1. parallelogram

  2. rhombus

  3. trapezium

  4. kite
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let angles be $3x, 7x, 6x$ and $4x$
Total sum of angles of a quadrilateral = $360^{\circ}$
$\Rightarrow 3x + 7x + 6x + 4x = 360^{\circ}$
$\Rightarrow 20x = 360^{\circ}$
$\Rightarrow x = 18^{\circ}$
$\therefore$ Angles are $ 3\, \times\, 18^{\circ} = 54^{\circ}$
 $ 7\, \times\, 18^{\circ} = 126^{\circ}$
 $ 6\, \times\, 18^{\circ} = 108^{\circ}$
 $ 4\, \times\, 18^{\circ} = 72^{\circ}$
All the angles of the figure ABCD are different, thus it is a trapezium.
Hence, option 'C' is correct.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

If the angles $A, B, C, D$ of a quadrilateral , taken in order are in the ratio $7:13:12:8$, then $ABCD$ is:

  1. rhombus

  2. parallelogram

  3. trapezium

  4. kite
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the angles be $7x, 13x, 12x$ and $8x$
Then, $7x+13x+12x+8x={360}^{o}$
$\Rightarrow$ $40x={360}^{o}$ $\Rightarrow$ $x={9}^{o}$
$\therefore$ $40x={360}^{o}$
$\therefore$ The angles taken in order are ${63}^{o}, {117}^{o}, {108}^{o}, {72}^{o}$ 
This shows that tow pairs of adjacent angles are supplementary $({63}^{o}+{117}^{o}={108}^{o}$ and ${108}^{o}+{72}^{o}={180}^{o}$), but opposite angles are not equal.
Therefore, the given quadrilateral will be a trapezium.

Multiple choice

Which of the following quadrilaterals has two congruent diagonals?

  1. Rectangle

  2. Square

  3. Rhombus

  4. Parallelogram
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A rhombus is a quadrilateral with two congruent diagonals.

Multiple choice

Which of the following quadrilaterals has four equal angles?

  1. Rectangle

  2. Square

  3. Rhombus

  4. Parallelogram
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A square is a quadrilateral with four equal angles.

Multiple choice

Which of the following quadrilaterals has four congruent sides and four right angles?

  1. Rectangle

  2. Square

  3. Rhombus

  4. Parallelogram
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A square is a quadrilateral with four congruent sides and four right angles.

Multiple choice

The parallelogram law in an inner product space states that:

  1. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 2\lVert x \rVert^2 + 2\lVert y \rVert^2$
  2. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = \lVert x \rVert^2 + \lVert y \rVert^2$
  3. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = \lVert x \rVert^2 - \lVert y \rVert^2$
  4. $\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 4\lVert x \rVert^2 + 4\lVert y \rVert^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The parallelogram law in an inner product space states that (\lVert x + y \rVert^2 + \lVert x - y \rVert^2 = 2\lVert x \rVert^2 + 2\lVert y \rVert^2).

Multiple choice

What is the name of the theorem that states that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides?

  1. Pythagorean Theorem

  2. Triangle Inequality Theorem

  3. Angle Sum Theorem

  4. Brahmagupta's Identity
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Brahmagupta's Identity states that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides. This theorem is attributed to Indian mathematicians.

Multiple choice

In a parallelogram, opposite sides are:

  1. Congruent

  2. Parallel

  3. Perpendicular
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a parallelogram, opposite sides are congruent.

Multiple choice

What is the name of the theorem that states that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides?

  1. Pythagorean theorem

  2. Brahmagupta theorem

  3. Angle sum theorem

  4. Parallelogram law
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The parallelogram law states that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Multiple choice

What is the name of the theorem that states that the diagonals of a cyclic quadrilateral are perpendicular if and only if the opposite sides are equal?

  1. Brahmagupta's theorem

  2. Ptolemy's theorem

  3. Pythagoras' theorem

  4. Euler's theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's theorem states that the diagonals of a cyclic quadrilateral are perpendicular if and only if the opposite sides are equal.

Multiple choice

What is the name of the theorem that states that the sum of the squares of the two diagonals of a parallelogram is equal to the sum of the squares of its four sides?

  1. Pythagorean theorem

  2. Triangle inequality theorem

  3. Euler's theorem

  4. Brahmagupta's theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Brahmagupta's theorem states that the sum of the squares of the two diagonals of a parallelogram is equal to the sum of the squares of its four sides. This theorem is named after Brahmagupta, who first discovered it in the 7th century AD.

Multiple choice

What is the name of the theorem that states that the diagonals of a rhombus bisect each other at right angles?

  1. Heron's formula

  2. Pythagorean theorem

  3. Brahmagupta's theorem

  4. Rhombus diagonal theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The rhombus diagonal theorem is a result in geometry that establishes a relationship between the diagonals of a rhombus.