# Find the area of the square whose one pair of opposite vertices are (3,4) and (5,6).

Last updated date: 19th Mar 2023

•

Total views: 303.9k

•

Views today: 7.82k

Answer

Verified

303.9k+ views

Hint: Calculate the length of diagonal of the square using distance formula between two points. Use the formula for calculating the area of square when the length of diagonal is given, which is \[\dfrac{{{a}^{2}}}{2}\] where ‘a’ is the length of the diagonal. You can also calculate the length of the side of the square when a diagonal is given and then calculate the area of the square.

Complete step by step answer:

We have the coordinates (3,4) and (5,6) of two opposite vertices of the square. We have to find the area of the given square. To find the area, we will firstly calculate the length of the diagonal of the square using distance formula between two points.

We know that distance between two points of the form \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] is \[\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}\].

Substituting \[{{x}_{1}}=3,{{y}_{1}}=4,{{x}_{2}}=5,{{y}_{2}}=6\] in the above formula, the distance between the points (3,4) and (5,6) is \[\sqrt{{{\left( 3-5 \right)}^{2}}+{{\left( 4-6 \right)}^{2}}}=\sqrt{{{2}^{2}}+{{2}^{2}}}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}\] units.

We will now evaluate the area of the square using the length of the diagonal of the square. We can also calculate the area of the square by finding the length of the edge of the square.

We know that the area of a square when the length of the edge is ‘a’ is \[{{a}^{2}}\]. So, the length of the diagonal is \[\sqrt{2}a\].

If we substitute \[x=\sqrt{2}a\], the area of the square will be \[\dfrac{{{x}^{2}}}{2}\].

Substituting \[x=2\sqrt{2}\], the area of square \[=\dfrac{{{a}^{2}}}{2}=\dfrac{{{\left( 2\sqrt{2} \right)}^{2}}}{2}=\dfrac{8}{2}=4\] sq. units.

Hence, the area of the square whose two opposite vertices are (3,4) and (5,6) is 4 sq. units.

Note: We can also solve this question by finding the length of the edge of the square using the length of diagonal of the square and then using it to calculate the area of the square. We can also solve this question by finding other two coordinates of the square and then finding the area of the square using the formula for calculating the area of a polygon when vertices are given.

Complete step by step answer:

We have the coordinates (3,4) and (5,6) of two opposite vertices of the square. We have to find the area of the given square. To find the area, we will firstly calculate the length of the diagonal of the square using distance formula between two points.

We know that distance between two points of the form \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] is \[\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}\].

Substituting \[{{x}_{1}}=3,{{y}_{1}}=4,{{x}_{2}}=5,{{y}_{2}}=6\] in the above formula, the distance between the points (3,4) and (5,6) is \[\sqrt{{{\left( 3-5 \right)}^{2}}+{{\left( 4-6 \right)}^{2}}}=\sqrt{{{2}^{2}}+{{2}^{2}}}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}\] units.

We will now evaluate the area of the square using the length of the diagonal of the square. We can also calculate the area of the square by finding the length of the edge of the square.

We know that the area of a square when the length of the edge is ‘a’ is \[{{a}^{2}}\]. So, the length of the diagonal is \[\sqrt{2}a\].

If we substitute \[x=\sqrt{2}a\], the area of the square will be \[\dfrac{{{x}^{2}}}{2}\].

Substituting \[x=2\sqrt{2}\], the area of square \[=\dfrac{{{a}^{2}}}{2}=\dfrac{{{\left( 2\sqrt{2} \right)}^{2}}}{2}=\dfrac{8}{2}=4\] sq. units.

Hence, the area of the square whose two opposite vertices are (3,4) and (5,6) is 4 sq. units.

Note: We can also solve this question by finding the length of the edge of the square using the length of diagonal of the square and then using it to calculate the area of the square. We can also solve this question by finding other two coordinates of the square and then finding the area of the square using the formula for calculating the area of a polygon when vertices are given.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

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

A Short Paragraph on our Country India