Math Concepts and Examples

May 14

Solve midpoint of quadrilateral article deals with the midpoint of sides of the quadrilateral, properties and the formula to solve the midpoints.

Midpoints of the quadrilaterals:

Quadrilaterals are two-dimensional closed four-sided figure with four midpoints on their sides. Each side of the quadrilateral has a midpoint. When the midpoint of the four sides are connected, it forms the parallelogram. This will form the quadrilateral inside a quadrilateral.

Various types of the shaped formed by connected the midpoints:

 

For solving the midpoints of the quadrilaterals, the following is very important

 Original quadrilateral                                       derived quadrilateral

  1. Rectangle                                                           Rhombus
  2. Square                                                                square
  3. Parallelogram                                                     parallelogram
  4. Rhombus                                                            Rectangle
  5. Trapezoid                                                            parallelogram
  6. isosceles trapezoid                                              Rhombus

Formula to solve midpoints of the quadrilaterals:

The formula

Midpoint = `((x1+x2)/2, (y1+y2)/2)`

Here (x1, y1) &(x2, y2) are points of the quadrilaterals.

Distance = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)` units.

Model problems to solve the midpoints of quadrilateral:

1. Find the shape formed by the connecting the midpoints of the quadrilateral whose vertices’s coordinates are (1, 2), (5, 2), (2, 6) and (6, 6).  

parallelogram midpoint

parallelogram midpoint                                                                        

Solution:

                The vertices’s coordinate of A (1, 2), B (7, 2), C (9, 6) and D (3, 6)

Formula for midpoint:

Midpoint = `((x1+x2)/2, (y1+y2)/2)`

Midpoint of AB = `(8/2, 4/2)`

P = (4, 2).

Midpoint of BC = `(16/2, 8/2)`

Q = (8, 4)

Midpoint of CD= `(12/2, 12/2)`

R = (6, 6)

Midpoint of AD = `(4/2, 8/2)`

S = (2, 4)

Since we have found the midpoints of the sides of the quadrilateral (rectangle). In next step, we can find the shape by connecting the midpoints of the sides.

Length of the PQ = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`                here (x1,y1)=(4,2)

= `sqrt ((8-4) ^2+ (4-2) ^2)`                              (x2, y2) = (8, 4)

= `sqrt (16+4)`

PQ = 4.5 units

Length of the QR = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`                here (x1, y1) = (8, 4)

= `sqrt ((6-8) ^2+ (6-4) ^2)`                                 (x2, y2) = (6, 6)

= `sqrt (4+4)`

QR= 2.8   units

Length of the RS = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`               here (x1,y1)=(6,6)

= `sqrt ((2-6) ^2+ (4-6) ^2)`                                 (x2,y2)=(2,4)

= `sqrt (16+4)`

RS = 4.5 units

Length of the SP = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`                here (x1,y1)=(2,4)

= `sqrt ((4-2) ^2+ (2-4) ^2)`                                 (x2,y2)=(4,2)

= `sqrt (4+4)`

SP = 2.8 units

Diagonal’s length:

Length of the PR = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`               here (x1,y1)=(4,2)

= `sqrt ((6-4) ^2+ (6-2) ^2)`                                 (x2,y2)=(6,6)

= `sqrt (20)`

PR = 4.5units

Length of the SQ = `sqrt ((x2-x1) ^2+ (y2-y1) ^2)`               here (x1,y1)=(2,4)

= `sqrt ((8-2) ^2+ (4-4) ^2)`                                 (x2,y2)=(8,4)

= `sqrt (36)`

SQ = 6 units

Here the opposite sides are equal in length and the diagonal are not equal in length.

Answer: the shape formed by the connecting the midpoint of the quadrilateral (parallelogram) is parallelogram.

From this example, the midpoint of the quadrilateral is learned.

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