Quadrilaterals MCQ Quiz - Objective Question with Answer for Quadrilaterals - Download Free PDF

Calculation:

Given,

Points A(1, 2, -1), B(2, 5, -2), and C(4, 4, -3) are three vertices of the rectangle.

We need to find the area of the rectangle formed by the vectors and .

Length of vector :

Length of vector :

Area of the rectangle is the product of the lengths of vectors and :

∴ The area of the rectangle is  square units.

Hence, the correct answer is Option 3.

Concept:

Area of Tetrahedron from Areas of Perpendicular Triangles:

• In a tetrahedron where edges AB, AC, and AD are mutually perpendicular, the area of the face opposite to vertex A (i.e., triangle BCD) is related to the areas of the triangles ABC, ACD, and ADB.
• These three triangles form right-angled faces and their areas can be used to find the area of the face BCD using Pythagorean relation in three dimensions.

The relation for the area of triangle BCD is,

Calculation:

Given,

⇒ 

∴ The area of triangle BCD is , hence the correct answer is Option 3.

Explanation:

Since diagonals of isosceles trapezium bisect each other, then the intersecting point of diagonal is O

Coordinates of O is 

∴ Option (c) is correct

Explanation:

In isosceles trapezium, diagonals bisect each other.

O is the midpoint of AC and BD

Then, 

⇒ a = 3, b = 1 

⇒ D(a, b) = (3, 1)

∴ Option (d) is correct

Explanation:

O is midpoint of P and R

⇒ O =  = (6,9)

Also, O is midpoint of S and Q.

⇒ O = 

⇒ (6,9) = 

Comparing both sides, we get

⇒ 

⇒x =4

Also 

⇒ 

⇒ y = 6

Thus, x + y = 4 + 6 = 10 

∴ Option (b) is correct

Concept:

ABCD is a parallelogram P, Q, R and S are mid-points of AB, BC, CD and DA respectively. Then PR and SQ bisect each other.

Calculation:

Applying the mid-point theorem,

⇒ x = 2

⇒ y = 10

So, the 4th coordinate is (2, 10)

Concept:

In triangle ABC, with sides a, b, and c sine rule is given by, 

sin (180 - θ) = sin θ 

Calculation:

1. In triangle ABD, using sine rule

sin θ /AB = sin α / AD

⇒ AD sin θ = AB sin α

So ,this is correct statement.

2.

In triangle ABD, using sine rule

sin θ /AB = sin (180-(θ +α )) / BD

⇒ BD sin θ = AB sin (α + θ)

So ,this is also correct statement.

Hence, option (3) is correct.

Concept:

Midpoint of () and () is given by, M = ()

Calculation:

Let, S(x, y, z) be the required point. Then  the midpoint of diagonal QS is 

()

And the midpoint of PR is 

(), i.e., ()

But the midpoint of diagonals of parallelogram coincide.

∴

So, x = -12, y = 0, and z = 5

Required point is S(-12, 0, 5)

Hence, option (4) is correct. 

Formula used:

The midpoint of two-point (x1, y1) and (x2, y2) is given by

Calculation:

Given that, point (2, 1) and (4, 3) are opposite vertices of rectangle 

The mid-point of A(2, 1) and C(4, 3) 

=  = (3, 2)

We know that the intersecting point of the diagonal of a rectangle is the same or at the midpoint. 

Therefore, the point (3, 2) will satisfy the equation 2x - y = k

⇒ 2(3) - 2 = k

⇒ k = 4

Hence, option 3 is correct.

Additional Information1. Equation of line of slope m passing through the point (x₁, y₁) is

(y - y₁) = m(x - x₁)

2. Equation of line passing through (x1, y1) and (x2, y2) is 

Concept:

Parallelogram:

• Opposite sides and opposite angles are equal.
• Digonals are not equal 

Rectangle:

• Rectangle is a parllelogram in which every angle is 90°

• Digonals are equal in lengths 

Rhombus: 

• It is a parallelogram in which all sides are equal
• Digonals are not equal in lengths

Square:

• All sides are equal and each angle is 90° 
• Diagonals are equal in lengths.

Distance between two points (x₁, y₁) and (x₂, y₂) is given by, 

Calculation:

Let, the given points be A(0, 5), B(-2, -2), C(5, 0) and D(7, 7). Then ABCD is a quadrilateral in which:

AB, BC, CD, and DA are sides

AB =  units

BC =  units

CD =  units

DA =  units

Here, AB = BC = CD = DA , i.e., all sides are equal.

Now, AC and BD are diagonals of ABCD

 units

And,  units

∴ AC ≠ BD ⇒ Digonals are not equal in length.

In ABCD, all sides are equal but digonals are not equal.

Thus, ABCD is a rhombus.

Hence, option (3) is correct.

Given:

 , 

Concept:

The area of parallelogram is

Calculation:

We have,

,  

Then

The area of parallelogram

Now

Then the area is 

= √ 14 sq unit

Hence the option (4) is correct.

CONCEPT:

Let A (x1, y1), B (x2, y2) and C (x3, y3) be the vertices of a Δ ABC, then area of Δ ABC = 

CALCULATION:

Given: A (0, 0), B (8, 0), C (3, 4), D (11, 4) are the vertices of a quadilateral ABCD

Here, we have to find the area of quadilateral ABCD

Area of quadilateral ABCD = Area of ΔABC + Area of Δ ACD

Let's find out the area of ΔABC

∵ A (0, 0), B (8, 0), C (3, 4) are the vertices of ΔABC

As we know that, if A (x1, y1), B (x2, y2) and C (x3, y3) be the vertices of a Δ ABC, then area of Δ ABC = 

⇒ Area of Δ ABC = 

Concept:

The equation of line is:

y = mx + c

where m is the slope of the line.

In the square, the diagonals are perpendicular to each other.

Calculations:

Given,

3x + 2y = 5 is the equation of one diagonal of the square.

Consider m₁ be the slope of one diagonal. and m₂ be the slope of other diagonal.

⇒ 2y = - 3x +5

⇒ 

⇒ m₁ = 

In square, the diagonals are perpendicular to each other.

⇒ m₁.m₂ = -1

⇒m₂ = 

Equation of other diagonal is 

 y = m₂x + c

⇒ y =  x + c....(1)

Since, the other diagonal is passing through the point (1, -1). It must satisfy the equation of line.

Put y = -1 and x = 1 in above equation

⇒ -1 =  + c

⇒ c = 

 Put the value of c in equation (1).

⇒y =  x -  

⇒ 3y = 2x - 5

⇒ 2x - 3y = 5 is the equation of other diagonal.

The point (1, -1) is one of the vertices of a square. If 3x + 2y = 5 is the equation of one diagonal of the square, then the equation of the other diagonal is 2x - 3y = 5

Concept:

The diagonals of the parallelogram bisect each other.

In a triangle PQR,  ​ 

For any vector 

Calculation:

Given:   and , 

Let the diagonals of the parallelogram intersect each other at O.

⇒   __(1)

(∵ The diagonals of the parallelogram bisect each other.)

As,   and , 

⇒   and , __(2)

From (1) and (2),

⇒   __(3)

Now in triangle PQO,

 +  +  = 0

⇒ 

⇒ 

Putting the values from equation (3),

⇒ 

⇒  = 

∴ The correct option. is (4) .

Concept:

Properties of a parallelogram

• The diagonals of a parallelogram bisect each other.

• Opposite sides of a parallelogram are congruent.

• Opposite angles of a parallelogram are congruent.

Calculation:

Diagonals of a parallelogram bisect each other,

Let the point of intersection of diagonals be P(x, y)

So, P is mid-point of AC,

Let the Fourth vertex be D (x_D, y_D)

P is mid-point of BD,

Fourth vertex of the parallelogram is D = (1, 2)