Answer

Verified

453.6k+ views

Hint: First name the coordinates of all the vertices of the parallelogram. Find the coordinates of mid-points of the diagonal using the coordinates of the vertices. Since it’s a parallelogram, coordinates bisect each other, hence, equate both coordinates of midpoint obtained from two diagonals and then proceed.

Complete step-by-step answer:

Let A(1,2), B(4,y), C(x,6) and D(3,5) are the vertices of a parallelogram ABCD

AC and BD are the diagonals.

O is the mid-point of AC and BD

The coordinates of mid-point are given by \[\left[ {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right]\]

If O is the mid - point of AC, then the coordinates of O are =\[\left( {\dfrac{{1 + x}}{2},\dfrac{{2 + 6}}{2}} \right) = \left( {\dfrac{{x + 1}}{2},4} \right)\]

If O is the mid-point of BD then coordinates of O are=\[\left( {\dfrac{{4 + 3}}{2},\dfrac{{5 + y}}{2}} \right) = \left( {\dfrac{7}{2},\dfrac{{5 + y}}{2}} \right)\]

Since both coordinates are of the same point O

∴ \[\dfrac{{1 + x}}{2} = \dfrac{7}{2}\]

\[ \Rightarrow 1 + x = 7\]

\[x = 7 - 1 = 6\]

∴\[\dfrac{{5 + y}}{2} = 4\]

\[ \Rightarrow 5 + y = 8\]

\[ \Rightarrow y = 8 - 5 = 3\]

Hence, x = 6 and y = 3.

Note: Following are the properties of Parallelogram are:

Opposite sides are parallel.

Opposite sides are congruent.

Opposite angles are congruent.

Consecutive angles are supplementary.

Diagonals intersect each other.

Complete step-by-step answer:

Let A(1,2), B(4,y), C(x,6) and D(3,5) are the vertices of a parallelogram ABCD

AC and BD are the diagonals.

O is the mid-point of AC and BD

The coordinates of mid-point are given by \[\left[ {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right]\]

If O is the mid - point of AC, then the coordinates of O are =\[\left( {\dfrac{{1 + x}}{2},\dfrac{{2 + 6}}{2}} \right) = \left( {\dfrac{{x + 1}}{2},4} \right)\]

If O is the mid-point of BD then coordinates of O are=\[\left( {\dfrac{{4 + 3}}{2},\dfrac{{5 + y}}{2}} \right) = \left( {\dfrac{7}{2},\dfrac{{5 + y}}{2}} \right)\]

Since both coordinates are of the same point O

∴ \[\dfrac{{1 + x}}{2} = \dfrac{7}{2}\]

\[ \Rightarrow 1 + x = 7\]

\[x = 7 - 1 = 6\]

∴\[\dfrac{{5 + y}}{2} = 4\]

\[ \Rightarrow 5 + y = 8\]

\[ \Rightarrow y = 8 - 5 = 3\]

Hence, x = 6 and y = 3.

Note: Following are the properties of Parallelogram are:

Opposite sides are parallel.

Opposite sides are congruent.

Opposite angles are congruent.

Consecutive angles are supplementary.

Diagonals intersect each other.

Recently Updated Pages

Which of the following is correct regarding the Indian class 10 social science CBSE

Who was the first sultan of delhi to issue regular class 10 social science CBSE

The Nagarjuna Sagar project was constructed on the class 10 social science CBSE

Which one of the following countries is the largest class 10 social science CBSE

What is Biosphere class 10 social science CBSE

Read the following statement and choose the best possible class 10 social science CBSE

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

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

Difference Between Plant Cell and Animal Cell

Why is the Earth called a unique planet class 6 social science CBSE