Q. 63 Use vector methods to show that ... [FREE SOLUTION]

Chapter 10: Q. 63 (page 802)

Use vector methods to show that the diagonals of a parallelogram bisect each other.

Short Answer

Expert verified

It is proved that the diagonals of a parallelogram bisect each other.

Step by step solution

Given

Let us consider a parallelogram $A B C D$ , $A$ be at origin and $\overset{鈫�}{A B} = \overset{鈫�}{a}, \overset{鈫�}{A D} = \overset{鈫�}{b}$

We have to prove that $H$ is the mid-point of $A C and B D$ .

Proof

Let us consider,

$\overset{鈫�}{A H} = x \overset{鈫�}{A C} and \overset{鈫�}{H B} = y \overset{鈫�}{D B} . . . (1)$

From above figure,

$\overset{鈫�}{A C} = \overset{鈫�}{A B} + \overset{鈫�}{B C} = \overset{鈫�}{A B} + \overset{鈫�}{A D} = \overset{鈫�}{a} + \overset{鈫�}{b} \overset{鈫�}{D B} = \overset{鈫�}{D A} + \overset{鈫�}{A B} = \overset{鈫�}{A B} - \overset{鈫�}{A D} = \overset{鈫�}{a} - \overset{鈫�}{b} From equation (1), \overset{鈫�}{A H} = x (\overset{鈫�}{a} + \overset{鈫�}{b}) \overset{鈫�}{H B} = y (\overset{鈫�}{a} - \overset{鈫�}{b}) Now, \overset{鈫�}{A B} = \overset{鈫�}{A H} + \overset{鈫�}{H B} 鈬� \overset{鈫�}{a} = x (\overset{鈫�}{a} + \overset{鈫�}{b}) + y (\overset{鈫�}{a} - \overset{鈫�}{b}) 鈬� \overset{鈫�}{a} = x \overset{鈫�}{a} + x \overset{鈫�}{b} + y \overset{鈫�}{a} - y \overset{鈫�}{b} 鈬� \overset{鈫�}{a} = (x + y) \overset{鈫�}{a} + (x - y) \overset{鈫�}{b}$

Equating the coefficient,

$x + y = 1 and x - y = 0$

After solving,

$x = \frac{1}{2} and y = \frac{1}{2}$

Therefore,

$\overset{鈫�}{A H} = \frac{1}{2} \overset{鈫�}{A C} and \overset{鈫�}{H B} = \frac{1}{2} \overset{鈫�}{D B}$

Hence, proved.

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91影视

Use vector methods to show that the diagonals of a parallelogram bisect each other.

Short Answer

Step by step solution

Given

Proof

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