
Prove that and 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 , where and be the two points.
Complete Complete Step by Step Solution:
We will first draw the diagrams showing all the points.
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 and B using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
…………………..
Now, we have to find the distance between B and C using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
……………….
Now, we have to find the distance between C and D using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
…………………….
Now, we have to find the distance between D and A using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
………………….
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 and C using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
………………
Now, we have to find the length of the diagonals B and D using the distance formula.
Substituting , , and in the formula , we get
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
………………….
Since the length of the diagonals are not equal. Hence the given points do not form a square.
Therefore, and 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.
Formula used:
We will use the formula of distance between two points which is given
Complete Complete Step by Step Solution:
We will first draw the diagrams showing all the points.

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
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
Now, we have to find the distance between B
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
Now, we have to find the distance between C
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
Now, we have to find the distance between D
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
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
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
Now, we have to find the length of the diagonals B
Substituting
Subtracting the terms in the bracket, we get
Applying the exponent on the terms, we get
Adding the terms, we get
Since the length of the diagonals are not equal. Hence the given points do not form a square.
Therefore,
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.
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