Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Questions > Four points P(0, 1), Q(0, –3), R(–2, –1) and...
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Four points P(0, 1), Q(0, –3), R(–2, –1) and S(2 , –1) represent the vertices of a quadrilateral. What is the area enclosed by the quadrilateral?
- a)8√2
- b)4
- c)8
- d)4√2
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Four points P(0, 1), Q(0, –3), R(–2, –1) and S(2 , –1) represent the ...
Given:
Four points P(0, 1), Q(0, –3), R(–2, –1), and S(2 , –1) represent the vertices of a quadrilateral.
To find:
The area enclosed by the quadrilateral.
Solution:
To find the area of the quadrilateral, we can divide it into two triangles and calculate the area of each triangle separately.
Triangle PQR:
Let's consider triangle PQR. The coordinates of the vertices are P(0, 1), Q(0, –3), and R(–2, –1).
- Length PQ: Using the distance formula, we can find the length of PQ as follows:
PQ = √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 0)^2 + (-3 - 1)^2)
= √(0 + 16)
= √16
= 4
- Length QR: Using the distance formula, we can find the length of QR as follows:
QR = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 0)^2 + (-1 - (-3))^2)
= √((-2)^2 + (2)^2)
= √(4 + 4)
= √8
= 2√2
- Length PR: Using the distance formula, we can find the length of PR as follows:
PR = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 0)^2 + (-1 - 1)^2)
= √((-2)^2 + (-2)^2)
= √(4 + 4)
= √8
= 2√2
- Using Heron's formula, we can calculate the area of triangle PQR as:
Area = √(s(s - PQ)(s - QR)(s - PR))
where s is the semi-perimeter of the triangle given by s = (PQ + QR + PR)/2.
Substituting the values, we get:
s = (4 + 2√2 + 2√2)/2
= (4 + 4√2)/2
= 2 + 2√2
Area = √((2 + 2√2)((2 + 2√2) - 4)((2 + 2√2) - 2√2)((2 + 2√2) - 2√2))
= √((2 + 2√2)(2 - 2)(2 - 2)(2 - 2))
= √(0)
= 0
Triangle RSQ:
Let's consider triangle RSQ. The coordinates of the vertices are R(–2, –1), S(2, –1), and Q(0, –3).
- Length RS: Using the distance formula, we can find the length of RS as follows:
RS = √((x
Four points P(0, 1), Q(0, –3), R(–2, –1), and S(2 , –1) represent the vertices of a quadrilateral.
To find:
The area enclosed by the quadrilateral.
Solution:
To find the area of the quadrilateral, we can divide it into two triangles and calculate the area of each triangle separately.
Triangle PQR:
Let's consider triangle PQR. The coordinates of the vertices are P(0, 1), Q(0, –3), and R(–2, –1).
- Length PQ: Using the distance formula, we can find the length of PQ as follows:
PQ = √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 0)^2 + (-3 - 1)^2)
= √(0 + 16)
= √16
= 4
- Length QR: Using the distance formula, we can find the length of QR as follows:
QR = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 0)^2 + (-1 - (-3))^2)
= √((-2)^2 + (2)^2)
= √(4 + 4)
= √8
= 2√2
- Length PR: Using the distance formula, we can find the length of PR as follows:
PR = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-2 - 0)^2 + (-1 - 1)^2)
= √((-2)^2 + (-2)^2)
= √(4 + 4)
= √8
= 2√2
- Using Heron's formula, we can calculate the area of triangle PQR as:
Area = √(s(s - PQ)(s - QR)(s - PR))
where s is the semi-perimeter of the triangle given by s = (PQ + QR + PR)/2.
Substituting the values, we get:
s = (4 + 2√2 + 2√2)/2
= (4 + 4√2)/2
= 2 + 2√2
Area = √((2 + 2√2)((2 + 2√2) - 4)((2 + 2√2) - 2√2)((2 + 2√2) - 2√2))
= √((2 + 2√2)(2 - 2)(2 - 2)(2 - 2))
= √(0)
= 0
Triangle RSQ:
Let's consider triangle RSQ. The coordinates of the vertices are R(–2, –1), S(2, –1), and Q(0, –3).
- Length RS: Using the distance formula, we can find the length of RS as follows:
RS = √((x
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Community Answer
Four points P(0, 1), Q(0, –3), R(–2, –1) and S(2 , –1) represent the ...
Given : The vertices of a Quadrilateral have four points P(0, 1), Q(0, –3), R(–2, –1) and S(2 , –1).
On plotting points on graph,
According to the given data a Quadrilateral PSQR will form as shown in below figure,
The distance formula between two points = 
We can see clearly,
Length of diagonal 
So, Quadrilateral PSQR follows property of square.
Area of Quadrilateral PSQR = Side2
Area of Quadrilateral PSQR = (√8)2 = 8 Sq. unit
and also by,
Area of Rhombus = 1/2(product of both diagonals)
= 1/2(4 x 4) Sq. unit = 8 Sq. unit
Hence, the correct option is (C).
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