
Prove that the points (-4, -1), (-2, -4), (4, 0) and (2, 3) are the vertices of a rectangle.
Answer
509.4k+ views
Hint: Firstly, we will find the lengths of all the sides of the rectangle and show that opposite sides are equal and then find the slopes and show that opposite sides are parallel. Thus, we have a parallelogram. Now, we just need to show two adjacent sides are perpendicular to each other.
Complete step-by-step answer:
To prove that a quadrilateral is a rectangle we need to show that opposite sides are equal and parallel and it has at least one right angle.
Let A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) be the given vertices. Now we need to show that ABCD is a rectangle.
First, let us find the side lengths of AB, BC, CD, AD.
We know that distance between two points (a, b) and (c, d) is given by the formula:
We have A(-4, -1) and B(-2, -4).
So,
……(1)
We have B(-2, -4) and C(4, 0).
So,
……….(2)
We have C(4, 0) and D(2, 3).
So,
……….(3)
We have A(-4, -1) and D(2, 3).
So,
……….(4)
Now comparing (1) and (3), we have:- AB = CD.
And comparing (2) and (4), we have:- BC = AD.
Hence, opposite sides are equal in length.
Now, we will find the slopes of the all the sides:-
Slope of line formed by two points (a, b) and (c, d) is given by .
We have A(-4, -1) and B(-2, -4).
So, Slope of ……..(5)
We have B(-2, -4) and C(4, 0).
So, Slope of ………(6)
We have C(4, 0) and D(2, 3).
So, Slope of ………(7)
We have D(2, 3) and A(-4, -1).
So, Slope of ………(8)
Now comparing (5) and (7), we have:- Slope of AB = Slope of CD.
And comparing (6) and (8), we have:- Slope of BC = Slope of AD.
Hence, opposite sides are parallel.
Hence, now, ABCD is a parallelogram.
(Because If a quadrilateral has equal opposite sides and parallel, then it is a parallelogram)
Now we can see from (5) and (6) that Slope of AB is a negative reciprocal of Slope of BC.
This happens when lines are perpendicular.
Similarly with (7) and (8).
Hence, we got a rectangle by the requirements mentioned in the start of the answer.
Note: The students might make the mistake of forgetting any one of the conditions of proving a quadrilateral is a rectangle like maybe he/she forgets to prove that opposite sides are parallel or equal.
Or he might stop at proving ABCD a parallelogram, but we need one right angle for sure.
Complete step-by-step answer:
To prove that a quadrilateral is a rectangle we need to show that opposite sides are equal and parallel and it has at least one right angle.
Let A(-4, -1), B(-2, -4), C(4, 0) and D(2, 3) be the given vertices. Now we need to show that ABCD is a rectangle.
First, let us find the side lengths of AB, BC, CD, AD.
We know that distance between two points (a, b) and (c, d) is given by the formula:
We have A(-4, -1) and B(-2, -4).
So,
We have B(-2, -4) and C(4, 0).
So,
We have C(4, 0) and D(2, 3).
So,
We have A(-4, -1) and D(2, 3).
So,
Now comparing (1) and (3), we have:- AB = CD.
And comparing (2) and (4), we have:- BC = AD.
Hence, opposite sides are equal in length.
Now, we will find the slopes of the all the sides:-
Slope of line formed by two points (a, b) and (c, d) is given by
We have A(-4, -1) and B(-2, -4).
So, Slope of
We have B(-2, -4) and C(4, 0).
So, Slope of
We have C(4, 0) and D(2, 3).
So, Slope of
We have D(2, 3) and A(-4, -1).
So, Slope of
Now comparing (5) and (7), we have:- Slope of AB = Slope of CD.
And comparing (6) and (8), we have:- Slope of BC = Slope of AD.
Hence, opposite sides are parallel.
Hence, now, ABCD is a parallelogram.
(Because If a quadrilateral has equal opposite sides and parallel, then it is a parallelogram)
Now we can see from (5) and (6) that Slope of AB is a negative reciprocal of Slope of BC.
This happens when lines are perpendicular.
Similarly with (7) and (8).
Hence, we got a rectangle by the requirements mentioned in the start of the answer.
Note: The students might make the mistake of forgetting any one of the conditions of proving a quadrilateral is a rectangle like maybe he/she forgets to prove that opposite sides are parallel or equal.
Or he might stop at proving ABCD a parallelogram, but we need one right angle for sure.
Recently Updated Pages
Master Class 10 Computer Science: Engaging Questions & Answers for Success

Master Class 10 Maths: Engaging Questions & Answers for Success

Master Class 10 English: Engaging Questions & Answers for Success

Master Class 10 General Knowledge: Engaging Questions & Answers for Success

Master Class 10 Science: Engaging Questions & Answers for Success

Master Class 10 Social Science: Engaging Questions & Answers for Success

Trending doubts
What is the past participle of wear Is it worn or class 10 english CBSE

What is Whales collective noun class 10 english CBSE

What is potential and actual resources

For what value of k is 3 a zero of the polynomial class 10 maths CBSE

What is the full form of POSCO class 10 social science CBSE

Which three causes led to the subsistence crisis in class 10 social science CBSE
