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Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.
Q. What is the value o f AC2 - BD2 ?
- a)25
- b)30
- c)36
- d)40
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6...
Suppose Mid point of AC and BD is M (a, b).
Most Upvoted Answer
Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6...
Solution:
To find the value of AC2 - BD2, we need to first find the length of AC and BD.
1. Length of AC:
AC is a diagonal of the parallelogram. We can use the distance formula to find the length of AC.
AC2 = (x - 1)2 + (6 - 2)2
AC2 = (x - 1)2 + 42
AC2 = x2 - 2x + 17
2. Length of BD:
BD is also a diagonal of the parallelogram. We can use the distance formula to find the length of BD.
BD2 = (3 - 4)2 + (5 - y)2
BD2 = 12 + (5 - y)2
BD2 = y2 - 10y + 34
Now, we can substitute these values in the expression AC2 - BD2.
AC2 - BD2 = (x2 - 2x + 17) - (y2 - 10y + 34)
AC2 - BD2 = x2 - 2x + 17 - y2 + 10y - 34
AC2 - BD2 = x2 - y2 - 2x + 10y - 17
We know that opposite sides of a parallelogram are equal in length. Therefore, we can use this property to find the values of x and y.
AB = DC
(4 - 1)2 + (y - 2)2 = (3 - x)2 + (5 - 6)2
9 + (y - 2)2 = (x - 3)2 + 1
(y - 2)2 - (x - 3)2 = -8
We also know that the diagonals of a parallelogram bisect each other. Therefore, we can use this property to find the values of x and y.
AC bisects BD
(1 + x)/2 = (3 + 4)/2
(2 + y)/2 = (5 + y)/2
Solving these equations, we get x = 5 and y = 4.
Substituting these values in the expression AC2 - BD2, we get:
AC2 - BD2 = 52 - 42 - 2(5) + 10(4) - 17
AC2 - BD2 = 25 - 16 - 10 + 40 - 17
AC2 - BD2 = 36
Therefore, the value of AC2 - BD2 is 36, which is option (c).
To find the value of AC2 - BD2, we need to first find the length of AC and BD.
1. Length of AC:
AC is a diagonal of the parallelogram. We can use the distance formula to find the length of AC.
AC2 = (x - 1)2 + (6 - 2)2
AC2 = (x - 1)2 + 42
AC2 = x2 - 2x + 17
2. Length of BD:
BD is also a diagonal of the parallelogram. We can use the distance formula to find the length of BD.
BD2 = (3 - 4)2 + (5 - y)2
BD2 = 12 + (5 - y)2
BD2 = y2 - 10y + 34
Now, we can substitute these values in the expression AC2 - BD2.
AC2 - BD2 = (x2 - 2x + 17) - (y2 - 10y + 34)
AC2 - BD2 = x2 - 2x + 17 - y2 + 10y - 34
AC2 - BD2 = x2 - y2 - 2x + 10y - 17
We know that opposite sides of a parallelogram are equal in length. Therefore, we can use this property to find the values of x and y.
AB = DC
(4 - 1)2 + (y - 2)2 = (3 - x)2 + (5 - 6)2
9 + (y - 2)2 = (x - 3)2 + 1
(y - 2)2 - (x - 3)2 = -8
We also know that the diagonals of a parallelogram bisect each other. Therefore, we can use this property to find the values of x and y.
AC bisects BD
(1 + x)/2 = (3 + 4)/2
(2 + y)/2 = (5 + y)/2
Solving these equations, we get x = 5 and y = 4.
Substituting these values in the expression AC2 - BD2, we get:
AC2 - BD2 = 52 - 42 - 2(5) + 10(4) - 17
AC2 - BD2 = 25 - 16 - 10 + 40 - 17
AC2 - BD2 = 36
Therefore, the value of AC2 - BD2 is 36, which is option (c).
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Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.Q. What is the value o f AC2 - BD2 ?a)25b)30c)36d)40Correct answer is option 'C'. Can you explain this answer?
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Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.Q. What is the value o f AC2 - BD2 ?a)25b)30c)36d)40Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.Q. What is the value o f AC2 - BD2 ?a)25b)30c)36d)40Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.Q. What is the value o f AC2 - BD2 ?a)25b)30c)36d)40Correct answer is option 'C'. Can you explain this answer?.
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