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Prove that (4,1),(6,0),(7,2) and (5,1) are the vertices of a rhombus. Is it a square?

Answer
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Hint: Here, we have to prove that the given points are the vertices of a rhombus. We will prove this by using the distance between two points formula. If all the length of the sides are equal then the given points will form the vertices of a rhombus. If the length of the diagonals are equal, then the vertices of a rhombus will form a square.

Formula used:
We will use the formula of distance between two points which is given d=(x2x1)2+(y2y1)2, where (x1,y1) and (x2,y2) be the two points.

Complete Complete Step by Step Solution:
We will first draw the diagrams showing all the points.
seo images

Let ABCD be the vertices of rhombus.
Now, we will be using the distance between two points formula for all the sides to prove that the given points are the vertices of a rhombus.
Now, we have to find the distance between A(4,1) and B (6,0) using the distance formula.
Substituting x1=4, x2=6, y1=1 and y2=0 in the formula d=(x2x1)2+(y2y1)2, we get
AB=(64)2+(0(1))2
Subtracting the terms in the bracket, we get
AB=(2)2+(1)2
Applying the exponent on the terms, we get
AB=4+1
Adding the terms, we get
AB=5 ………………….. (1)
Now, we have to find the distance between B(6,0) and C (7,2) using the distance formula.
Substituting x1=6, x2=7, y1=0 and y2=2 in the formula d=(x2x1)2+(y2y1)2, we get
BC=(76)2+(20)2
Subtracting the terms in the bracket, we get
BC=12+22
Applying the exponent on the terms, we get
BC=1+4
Adding the terms, we get
BC=5 ……………….(2)
Now, we have to find the distance between C (7,2) and D (5,1) using the distance formula.
Substituting x1=7, x2=5, y1=2 and y2=1 in the formula d=(x2x1)2+(y2y1)2, we get
CD=(57)2+(12)2
Subtracting the terms in the bracket, we get
CD=(2)2+(1)2
Applying the exponent on the terms, we get
CD=4+1
Adding the terms, we get
CD=5 ……………………. (3)
Now, we have to find the distance between D (5,1)and A(4,1)using the distance formula.
Substituting x1=7, x2=5, y1=2 and y2=1 in the formula d=(x2x1)2+(y2y1)2, we get
DA=(45)2+(11)2
Subtracting the terms in the bracket, we get
DA=(1)2+(2)2
Applying the exponent on the terms, we get
DA=1+4
Adding the terms, we get
DA=5 …………………. (4)
Since all the length of the sides of a square are equal, the given points form the vertices of a rhombus.
Now, we have to check whether it is a square.
Now, we have to find the length of the diagonals A (4,1) and C (7,2) using the distance formula.
Substituting x1=4, x2=7, y1=1 and y2=2 in the formula d=(x2x1)2+(y2y1)2, we get
AC=(74)2+(2(1))2
Subtracting the terms in the bracket, we get
AC=(3)2+(3)2
Applying the exponent on the terms, we get
AC=9+9
Adding the terms, we get
AC=18 ……………… (5)
Now, we have to find the length of the diagonals B (6,0) and D (5,1) using the distance formula.
Substituting x1=6, x2=5, y1=0 and y2=1 in the formula d=(x2x1)2+(y2y1)2, we get
BD=(56)2+(10)2
Subtracting the terms in the bracket, we get
BD=(1)2+(1)2
Applying the exponent on the terms, we get
BD=1+1
Adding the terms, we get
BD=2 ………………….(6)

Since the length of the diagonals are not equal. Hence the given points do not form a square.
Therefore, (4,1),(6,0),(7,2) and (5,1) are the vertices of a rhombus and it is not a square.


Note:
We can prove that the given vertices form a parallelogram by using the midpoint formula. The midpoint formula can be used for finding the midpoint of the diagonal. If the midpoint of both the diagonals are equal, then it forms a parallelogram. If the opposite sides of a parallelogram are equal, then it forms a rhombus. If the length of the diagonal are equal, then it forms a square.