
Name the type of quadrilateral formed if any, by the following points and give reasons for your answer.
i) (-1, -2), (1, 0), (-1, 2), (-3, 0)
ii) (-3, 5), (3, 1), (0, 3), (-1, -4)
iii) (4, 5), (7, 6), (4, 3), (1, 2)
Answer
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Hint: In this question, we will use the basic concept of distance formula between two consecutive points. If all the lengths are equal, then we will find the lengths of the alternate points because it can be a square or a rhombus. And if two alternate sides are equal in length, then there are possibilities that the quadrilateral may be either rectangle or parallelogram and so the diagonal lengths are calculated then.
Complete step-by-step answer:
In this question, we will use the basic concept of distance formula, that is distance between and . Let us consider each of the options given in the question one by one.
i) (-1, -2), (1, 0), (-1, 2), (-3, 0)
Let and . So, we will now calculate and . After that we will calculate the length of and , then decide the shape of the quadrilateral. So, we can write the distance between and . So, we get,
Similarly, for , we get,
And, for , we get,
And similarly, for , we get,
So, here we can see from equations (i), (ii), (iii) and (iv) that . So, we can say that the quadrilateral is either a square or a rhombus. Now, we will calculate the length of its diagonals, that is, and . So, by calculating the length for , we get,
Similarly, we will calculate the length for and we get,
We can see that the diagonals and earlier we had observed that .
Therefore, we can say that the quadrilateral formed by these set of points is nothing but a square.
ii) (-3, 5), (3, 1), (0, 3), (-1, -4)
Let and . So, we will now calculate and . After that we will calculate and , and with the lengths of the sides and the diagonals, we will decide the shape of the quadrilateral that is formed. So, we can write the distance between and . So, we get,
Similarly, for , we get,
And, for , we get,
And similarly, for , we get,
In this set of points, we can see from equations (i), (ii), (iii) and (iv) that .
Therefore, the quadrilateral that is formed by these set of points is not a uniform quadrilateral.
iii) (4, 5), (7, 6), (4, 3), (1, 2)
Let and . So, we will now calculate the lengths of and . After that, if needed, we will calculate the lengths of and , and decide the shape of the quadrilateral that is formed. So, we can write the distance between and . So, we get,
Similarly, for , we get,
And, for , we get,
And similarly, for , we get,
Here, we can observe from equations (i), (ii), (iii) and (iv) that and . So, we can say that the quadrilateral is either a rectangle or a parallelogram. So, we will calculate the length of the diagonals, that is and . So, by calculating the length for , we get,
Similarly, we will calculate the length for and we get,
WE can see here that and earlier we had found out that and . Therefore, the quadrilateral formed by these set of points is a parallelogram.
Note: In this question, we have the most basic coordinate geometry formula, that is distance formula, which is for two points, and . The possible mistakes one can make in this question is by not calculating the length of the diagonals and directly predicting the shape of the quadrilateral. One should also be careful with the numerical signs while doing the calculations.
Complete step-by-step answer:
In this question, we will use the basic concept of distance formula, that is distance between
i) (-1, -2), (1, 0), (-1, 2), (-3, 0)
Let
Similarly, for
And, for
And similarly, for
So, here we can see from equations (i), (ii), (iii) and (iv) that
Similarly, we will calculate the length for
We can see that the diagonals
Therefore, we can say that the quadrilateral formed by these set of points is nothing but a square.
ii) (-3, 5), (3, 1), (0, 3), (-1, -4)
Let
Similarly, for
And, for
And similarly, for
In this set of points, we can see from equations (i), (ii), (iii) and (iv) that
Therefore, the quadrilateral that is formed by these set of points is not a uniform quadrilateral.
iii) (4, 5), (7, 6), (4, 3), (1, 2)
Let
Similarly, for
And, for
And similarly, for
Here, we can observe from equations (i), (ii), (iii) and (iv) that
Similarly, we will calculate the length for
WE can see here that
Note: In this question, we have the most basic coordinate geometry formula, that is distance formula, which is for two points,
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