Explanation Given information : Statement of the theorem: Sum of squares of the lengths of the sides of a parallelogram equals the sum of the squares of the lengths of the diagonals. Formula used : The position vector of a point A (say) is a vector O A → originating from an arbitrary point O. The magnitude of vector O A → is proportional to the length of vector O A → denoted by | O A → | . Calculations: Let OABC is a parallelogram with sides OA ∥ CB and OC ∥ AB . Let the position vectors of points A and Cwith respect to origin O are a → and c → respectively. OB and ACare two diagonalsof the parallelogram shown in figure here. Now, O A → = a → ⇒ | O A → | = | a → | = O A , and O C → = c → ⇒ | O C → | = | c → | = A C As two parallel vectors p → and q → may be written as p → = λ q → , and when they are equal then λ=1 . Since, OA ∥ CB and OC ∥ AB Therefore, C B → = O A → = a → ⇒ | C B → | = | a → | = C B , and A B → = O C → = c → ⇒ | A B → | = | c → | = A B ( parallel and equal sides). Thus, we have O A = | a → | , O A = | a → | = C B , and O C = | c → | = A B ... (1) Now the vector of diagonal O B → = O C → + C B → = c → + a → ⇒ O B = | O B → | = | c → + a → |

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 9, Problem 61SP

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Chapter 9, Problem 60SP

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Toshow: The sum of squares of the lengths of the sides of a parallelogram equals the sum of the squares of the lengths of the diagonals.

Solution Summary: The author explains the statement of the theorem: Sum of squares equals the sum of diagonals.

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Toshow:The sum of squares of the lengths of the sides of a parallelogram equals the sum of the squares of the lengths of the diagonals.

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