Mensuration

Results of Quadrilaterals

One of the topics that are asked in all the competitive exams in quantitative aptitude is mensuration. Mensuration has many different kinds of topics in it. Out of these topics, the quadrilateral is one of them. And today we are going to discuss results of quadrilaterals in detail. We will also explain you the types of quadrilaterals and it’s properties. Later we will discuss some solved examples.

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Quadrilaterals

A polygon that has four sides is called a quadrilateral. Thus squares, rectangles, rhombus, trapezium are quadrilaterals.

The area of a quadrilateral = ½ × (product of diagonals) × (sine of the angle between them).

Results of Quadrilaterals

From the figure, you can see the Θ1 and Θ2 are sine angles between the diagonals. The total angle made by these angles is always 180°. So, it can be said that this two angles will always be supplementary to each other.

Browse more Topics under Mensuration

Properties of a Quadrilateral

  1. The total sum of the angles inside a quadrilateral is always 360°.
  2. For a convex quadrilateral, the sum of the product of sides is always equal to the product of the diagonals.

Types of Quadrilaterals

As mentioned above there are five types of the quadrilateral.

  1. Rhombus
  2. Square
  3. Rectangle
  4. Trapezium
  5. Parallelogram

Rhombus

A rhombus is a parallelogram that has all the sides equal. Area of a rhombus = 1/2 × d1 × d2 x sin90°

Properties:

  1. In a rhombus, diagonals always bisect each other.
  2. Every rhombus is parallelogram but all parallelogram may or may not be a rhombus.
  3. All squares are rhombus but vice versa may or may not be true.

Squares

A rhombus which has all the angles 90° or a rectangle with all the sides equal is called a square. Area of a square = height × base

Properties:

  1. In a square, the diagonals are of equal length and bisect each other at 90°.
  2. The diagonal of a square is the diameter of the circumscribed circle.
  3. The side of a square is the diameter of an inscribed circle.

Rectangle

A parallelogram which has all the angles of 90° is called a rectangle. Area of the rectangle = product of the two adjacent sides.

Properties:

  1. In the rectangle, the diagonals are equal and bisect each other.
  2. The angles that bisect a rectangle forms another rectangle.
  3. All rectangles are parallelogram but all the parallelograms may or may not be the rectangle.

Parallelogram

A quadrilateral whose opposite sides are parallel is called a parallelogram. Area of the parallelogram = base × height or area of the parallelogram = product of any two adjacent sides x sine of the included angle.

Properties:

  1. Diagonals of a parallelogram bisect each other.
  2. A parallelogram inside a circle is called rectangle.
  3. A parallelogram outside a circle is called rhombus.
  4. In a parallelogram the opposite angles are equal.
  5. The sum of the squares of the four sides is equal to the sum of the squares of the diagonals.

Trapezium

A quadrilateral whose only two sides are parallel to each other is called a trapezium. Area = ½ × sum of parallel sides × height.

Properties:

  1. If the non-parallel sides in the trapezium are equal then diagonals will be equal too.

Solved examples

Q. In a square ABCD, P is the mid-point of AB and Q is the mid-point of BC if the area of ∆PBQ is 100 m² than the area of the square ABCD =?

A. 250 m²           B. 400 m²          C. 800 m²              D. 600 m²

Area of ∆PBQ = ½ × PQ × QB = 100 = 100 × ½ × AB/2 x BC/2 = 100. PQ x QR = 2 × 2 × 2 × 100 = 800 cm². So, the correct answer is C.

Q. WXYZ is a quadrilateral and WX || YZ. T is the mid-point of WX. ZT || XY. If the area of the triangle ∆WTZ is 50 cm2 then the area of quadrilateral WXYZ is:?

A. 175 cm²        B. 100 cm²       C. 150 cm²         D. 125 cm²

Based on the data, this will be the figure:

Results of Quadrilaterals

WT = TX

Draw ZX perpendicular to WX, if ZX = h and WT = TX = a.

Area of ∆ WZT = ½ × a × h = ah/2.

Area of quadrilateral WXYZ = Area of ∆ WZT + Area of quadrilateral ZTXY

= ah/2 + ah

= 3ah/2

= 3(50)

= 150 cm2.

So the correct answer is C

Q. A triangle and a parallelogram are constructed on the same base such that their areas are equal. If the altitude of the parallelogram is 100 m, then the altitude of the triangle is:?

A. 200m        B. 50m           C. 100m            D. 125m

In this question, we will denote ‘b’ as the base hand h2 as the altitudes of the triangle and parallelogram respectively.

Then according to the data in the question:

½ × b ×  h1 = b ×  h2

h1 = 2 h2

h1 = 2  × 100 = 200m.

Thus the correct answer is A.

Practice Questions

Q. Quadrilateral ABCD is a parallelogram, AC, BD is the diagonals & intersect at point O. X and Y are the centroids of ∆ ADC and ∆ ABC respectively. If BY = 6 cm, then OX = ?

A. 3cm          B. 2cm           C. 4cm             D. 6cm

The correct answer is A.

Q. In a parallelogram ABCD if bisectors A and B meet at X, then the value of AXB is?

A. 45°          B. 90°        C. 75°             D. 60°

The correct answer is B.

Q. If one diagonal of a rhombus is equal to its side, then the diagonals of the rhombus are in the ratio:?

A. 2: 1          B. 3: 1             C. √3: 1            D. None of these

The correct answer is C.

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