To prove that the given points will create a parallelogram, we have to remember the following points.
Problem 1 :
Verify that the quadrilateral with vertices P(-2, 2), Q(-2, -3), R(-5, -5) and S(-5, 0) is a parallelogram.
Solution :
Slope of PQ :
P(-2, 2) and Q(-2, -3)
x_{1} = -2, y_{1} = 2, x_{2} = -2, y_{2} = -3
Slope of QR :
Q(-2, -3) and R(-5, -5)
x_{1} = -2, y_{1} = -3, x_{2} = -5, y2 = -5
Slope of RS :
R(-5, -5) and S(-5, 0)
x_{1} = -5, y_{1} = -5, x_{2} = -5, y_{2} = 0
Slope of SP :
S(-5, 0) and P(-2, 2)
x_{1} = -5, y_{1} = 0, x_{2} = -2, y_{2} = 2
The opposite sides PQ, RS and QR, SP have the same slope and are thus parallel to each other.
Midpoint of the diagonals :
Midpoint of diagonal PR :
P(-2, 2) and R(-5, -5)
Midpoint of S(-5, 0) and Q(-2, -3) :
Since the midpoints of the diagonal are equal and opposite sides are equal, it must be a parallelogram.
Problem 2 :
A quadrilateral has vertices K(-1, 4), L(2, 2), M(0, -1) and N(-3, 1). Verify that :
a) The quadrilateral is a square.
b) Each diagonal of the quadrilateral is the perpendicular bisector of the other diagonal.
c) The diagonals are equal in length.
Solution :
a) Given, K(-1, 4), L(2, 2), M(0, -1) and N(-3, 1)
Length of KL :
K(-1, 4) and L(2, 2)
x_{1} = -5, y_{1} = 0, x_{2} = -2, y_{2} = 2
KL = √[(-2 + 5)^{2} + (2 - 0)^{2}]
= √[(3)^{2} + (2)^{2}]
= √[9 + 4]
KL = √13
Length of LM :
L(2, 2) and M(0, -1)
x_{1} = 2, y_{1} = 2, x_{2} = 0 , y_{2} = -1
LM = √[(0 - 2)^{2} + (-1 - 2)^{2}]
= √[(-2)^{2} + (-3)^{2}]
= √[4 + 9]
LM = √13
Length of MN :
M(0, -1) and N(-3, 1)
x_{1} = 0, y_{1} = -1, x_{2} = -3 , y_{2} = 1
MN = √[(-3 - 0)^{2} + (1 + 1)^{2}]
= √[(-3)^{2} + (2)^{2}]
= √[9 + 4]
MN = √13
Length of NK :
N(-3, 1) and K(-1, 4)
x_{1} = -3, y_{1} = 1, x_{2} = -1, y_{2} = 4
NK = √[(-1 + 3)^{2} + (4 - 1)^{2}]
= √[(2)^{2} + (3)^{2}]
= √[4 + 9]
NK = √13
KL = LM = MN = NK
c) Length of KM :
K(-1, 4) and M(0, -1)
x_{1} = -1, y_{1} = 4, x_{2} = 0, y_{2} = -1
KM = √[(0 + 1)^{2} + (-1 - 4)^{2}]
= √[(1)^{2} + (-5)^{2}]
= √[1 + 25]
KM = √26
Length of NL :
N(-3, 1) and L(2, 2)
x_{1} = -3, y_{1} = 1, x_{2} = 2, y_{2} = 2
NL = √[(2 + 3)^{2} + (2 - 1)^{2}]
= √[(5)^{2} + (1)^{2}]
= √[25 + 1]
NL = √26
KM = NL
So, diagonals are equal.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM