ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer
Solution:
The distance between the two points can be measured using the Distance Formula which is given by:
Distance Formula = √( x₂ - x₁ )2 + (y₂ - y₁)2
From the figure given below,
P is the mid-point of side AB. The co-ordinates of P can be calculated using the Mid- Point Formula as,
M = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]
Considering A(-1, -1) and B(-1, 4), the coordinates of P are
= [(- 1 - 1) / 2, (- 1 + 4) / 2]
= (- 1, 3/2)
Similarly, the coordinates of Q, R, and S are calculated using the Mid-Point Formula as (2, 4), (5, 3/2), (2, - 1) respectively.
We know that the distance between the two points is given by the Distance Formula,
Distance Formula = √ [( x₂ - x₁ )2 + (y₂ - y₁)2].....(1)
Distance between two points P(- 1, 3/2) and Q(2, 4)
Length of PQ = √[(-1 - 2)2 + (3/2 - 4)2] = √(61/4) = √61/2
Distance between two points S(2, - 1) and P(- 1, 3/2)
Length of SP = √[(2 + 1)2 + (-1 - 3/2)2] = √(61/4) = √61/2
Distance between two points R(5, 3/2) and Q(2, 4)
Length of QR = √[(2 – 5)2 + (4 – 3/2)2] = √(61/4) = √61/2
Distance between two points R(5, 3/2) and S(2, - 1)
Length of RS = √[(5 – 2)2 + (3/2 + 1)2] = √(61/4) = √61/2
Distance between two points P(- 1, 3/2) and R(5, 3/2)
Length of PR (diagonal) = √[(-1 - 5)2 + (3/2 – 3/2)2] = 6
Distance between two points S(2, - 1) and Q(2, 4)
Length of QS (diagonal) = √[(2 – 2)2 + (4 + 1)2] = 5
Hence we can observe from the above values,
PQ = SP = QR = RS = √61/2, that is all sides are equal.
But PR is not equal to QS
which shows diagonals are not of equal measure.
Hence, it can be observed that all the given sides of the given quadrilateral are of equal measure.
However, the diagonals are of different lengths.
Therefore, PQRS is a rhombus.
☛ Check: NCERT Solutions Class 10 Maths Chapter 7
Video Solution:
ABCD is a rectangle formed by the points A(–1, –1), B(– 1, 4), C(5, 4) and D(5, – 1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus?
Maths NCERT Solutions Class 10 Chapter 7 Exercise 7.4 Question 8
Summary:
ABCD is a rectangle formed by the points A (- 1, - 1), B (- 1, 4), C (5, 4) and D (5, - 1). P, Q, R and S are the mid-points of AB, BC, CD, and DA respectively. It can be observed that all sides of the given quadrilateral are of the same measure. However, the diagonals are of different lengths. Therefore, PQRS is a rhombus.
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