(i) Find the length of all sides.
(ii) In a parallelogram, the length of opposite sides will be equal.
Question 1 :
A(4, 6), B(7, 7) C(10, 10) and D (7, 9)
Solution :
Distance Between Two Points (x ₁, y₁) and (x₂ , y₂)
√(x2 - x1)2 + (y2 - y1)2
Length of AB :
Here x1 = 4, y1 = 6, x2 = 7 and y2 = 7
= √(7-4)2 + (7-6)2
= √32+12
= √10
Length of BC :
Here x1 = 7, y1 = 7, x2 = 10 and y2 = 10
= √(10-7)2 + (10-7)2
= √32+32
= √(9+9)
= √18 units
Length of CD :
Here x1 = 10, y1 = 10, x2 = 7 and y2 = 9
= √(7-10)2 + (9-10)2
= √(-3)² + (-1)²
= √9 + 1
= √10 units
Length of DA :
Here x1 = 7, y1 = 9, x2 = 4 and y2 = 6
= √(4-7)² + (6-9)²
= √(-3)² + (-3)²
= √9 + 9
= √18 units
Length of opposite sides are equal. So, the given vertices will form a parallelogram.
Question 2 :
Examine whether the given points
A(3, -5), B(-5, -4), C (7, 10) and D (15, 9)
forms a parallelogram.
Solution :
Length of AB :
Here x1 = 3, y1 = -5, x2 = -5 and y2 = -4
= √(-5-3)2 + (-4-(-5))2
= √(-8)2+(-4+5)2
= √(64+1)
= √65 units
Length of BC :
Here x1 = -5, y1 = -4, x2 = 7 and y2 = 10
= √(7-(-5))2 + (10-(-4))2
= √(7+5)2 + (10+4)2
= √122 + 142
= √144 + 196
= √340 units
Length of CD :
Here x1 = 7, y1 = 10, x2 = 15 and y2 = 9
= √(15-7)2 + (9-10)2
= √82 + (-1)²
= √64 + 1
= √65 units
Length of DA :
Here x1 = 15, y1 = 9, x2 = 3 and y2 = -5
= √(3-15)² + (-5-9)²
= √(-12)² + (-14)²
= √144 + 196
= √340 units
Length of opposite sides are equal. So, the given vertices will form a parallelogram.
Question 3 :
Examine whether the given points
A (-4, -3) and B (3, 1) and C (3, 6) and D (-4, 2)
forms a parallelogram.
Solution :
Length of AB :
Here x1 = -4, y1 = -3, x2 = 3 and y2 = 1
= √(3-(-4))2+(1-(-3))2
= √(3+4)2+(1+3)2
= √72+42
= √49+16
= √65 units
Length of BC :
Here x1 = 3, y1 = 1, x2 = 3 and y2 = 6
= √(3-3)2 + (6-1)2
= √5²
= 5 units
Length of CD :
Here x1 = 3, y1 = 6, x2 = -4 and y2 = 2
= √(-4-3)2 + (2-6)2
= √(-7)2 + (-4)2
= √(49+16)
= √65 units
Length of DA :
Here x1 = -4, y1 = 2, x2 = -4 and y2 = -3
= √(-4-(-4))² + (-3-2)²
= √(-4+4)² + (-5)²
= √25
= 5 units
Length of opposite sides are equal. So, the given vertices will form a parallelogram.
Question 4 :
Examine whether the given points
A(8, 4), B(1, 3), C(3, -1) and D (4, 6)
forms a parallelogram.
Solution :
Length of AB :
Here x1 = 8, y1 = 4, x2 = 1 and y2 = 3
= √(1-8)² + (3-4)²
= √(-7)² + (-1)²
= √49 + 1
= √50 units
Length of BC :
Here x1 = 1, y1 = 3, x2 = 3 and y2 = -1
= √(3-1)² + (-1-3)²
= √(2)² + (-4)²
= √4 + 16
= √20 units
Length of CD :
Here x1 = 3, y1 = -1, x2 = 4 and y2 = 6
= √(4-3)² + (6-(-1))²
= √(1)² + (6+1)²
= √1 + 7²
= √1 + 49
= √50 units
Length of DA :
Here x1 = 4, y1 = 6, x2 = 8 and y2 = 4
= √(8-4)2 + (4-6)2
= √42 + (-2)2
= √16 + 4
= √20 units
Length of opposite sides are equal. So the given vertices forms a parallelogram
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