Answer

Verified

451.8k+ views

Hint: - Here, we made a quadrilateral whose one diagonal is x axis. And then suppose the coordinate of the corner of the quadrilateral with respect to x axis and y axis. And then go through the bisector formula of coordinates.

Let ABCD be the quadrilateral such that diagonal AC is along x axis suppose the coordinates A, B, C and D be \[(0,0),({x_2},{y_2})({x_1},0)\]and \[({x_3},{y_3})\] respectively.

E and F are the mid points of sides AD and BC respectively, G and H are the midpoint of diagonals AC and BD. And the point of intersection of EF and GH is I.

Coordinates of E are \[\left( {\dfrac{{0 + {x_3}}}{2},\dfrac{{0 + {y_3}}}{2}} \right) = \left( {\dfrac{{{x_3}}}{2},\dfrac{{{y_3}}}{2}} \right)\]

Coordinates of Fare\[\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{0 + {y_2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_2}}}{2}} \right)\]

Coordinates of midpoint of EF are \[\left( {\dfrac{{\dfrac{{{x_3}}}{2} + \dfrac{{{x_1} + {x_2}}}{2}}}{2},\dfrac{{\dfrac{{{y_3}}}{2} + \dfrac{{{y_2}}}{2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{4},\dfrac{{{y_2} + {y_3}}}{4}} \right)\]

G and H are the mid points of diagonal AC and BD respectively then

Coordinates of G are\[\left( {\dfrac{{0 + {x_1}}}{2},\dfrac{{0 + 0}}{2}} \right) = \left( {\dfrac{{{x_1}}}{2},0} \right)\]

Coordinates of H are\[\left( {\dfrac{{{x_2} + {x_3}}}{2},\dfrac{{{y_2} + {y_3}}}{2}} \right)\]

Coordinates of midpoint of GH are \[\left( {\dfrac{{\dfrac{{{x_1}}}{2} + \dfrac{{{x_2} + {x_3}}}{2}}}{2},\dfrac{{\dfrac{{{y_2}}}{2} + \dfrac{{{y_2} + {y_3}}}{2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{4},\dfrac{{{y_2} + {y_3}}}{4}} \right)\]

As you can see, midpoints of both EF and GH are the same. So, EF and GH meet and bisect each other.

Hence, proved.

Note:-Whenever we face such types of questions first of all make the diagram by the statement given by the question, then use the section formula to find the midpoint of a line. If the two points have the same value then it must coincide.

Let ABCD be the quadrilateral such that diagonal AC is along x axis suppose the coordinates A, B, C and D be \[(0,0),({x_2},{y_2})({x_1},0)\]and \[({x_3},{y_3})\] respectively.

E and F are the mid points of sides AD and BC respectively, G and H are the midpoint of diagonals AC and BD. And the point of intersection of EF and GH is I.

Coordinates of E are \[\left( {\dfrac{{0 + {x_3}}}{2},\dfrac{{0 + {y_3}}}{2}} \right) = \left( {\dfrac{{{x_3}}}{2},\dfrac{{{y_3}}}{2}} \right)\]

Coordinates of Fare\[\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{0 + {y_2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_2}}}{2}} \right)\]

Coordinates of midpoint of EF are \[\left( {\dfrac{{\dfrac{{{x_3}}}{2} + \dfrac{{{x_1} + {x_2}}}{2}}}{2},\dfrac{{\dfrac{{{y_3}}}{2} + \dfrac{{{y_2}}}{2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{4},\dfrac{{{y_2} + {y_3}}}{4}} \right)\]

G and H are the mid points of diagonal AC and BD respectively then

Coordinates of G are\[\left( {\dfrac{{0 + {x_1}}}{2},\dfrac{{0 + 0}}{2}} \right) = \left( {\dfrac{{{x_1}}}{2},0} \right)\]

Coordinates of H are\[\left( {\dfrac{{{x_2} + {x_3}}}{2},\dfrac{{{y_2} + {y_3}}}{2}} \right)\]

Coordinates of midpoint of GH are \[\left( {\dfrac{{\dfrac{{{x_1}}}{2} + \dfrac{{{x_2} + {x_3}}}{2}}}{2},\dfrac{{\dfrac{{{y_2}}}{2} + \dfrac{{{y_2} + {y_3}}}{2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{4},\dfrac{{{y_2} + {y_3}}}{4}} \right)\]

As you can see, midpoints of both EF and GH are the same. So, EF and GH meet and bisect each other.

Hence, proved.

Note:-Whenever we face such types of questions first of all make the diagram by the statement given by the question, then use the section formula to find the midpoint of a line. If the two points have the same value then it must coincide.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE