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Locus of the Points Equidistant From Two Intersecting Lines

40. If a hexa...

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40. If a hexagon PQRSTU circumscribes a circle, prove that PQ+RS+TU= QR+ST+UP

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Q. Assertion :Let the vectors

P Q, Q R, R S, S T, T U

U P

represent the sides the regular hexagon

P Q \times (R S + S T) \neq 0

P Q \times R S = 0

P Q \times S T \neq 0

Q. Assertion :Let the vectors

- - \to P Q

- - \to Q R

- - \to R S

- \to S T

- - \to T U

- - \to U P

represent the sides of a regular hexagon,

- - \to P Q \times (- - \to R S + - \to S T) \neq \to 0

- - \to P Q \times - - \to R S = \to 0, - - \to P Q \times - \to S T \neq 0

Q. In the figure, a quadrilateral

P Q R S

is drawn to circumscribe a circle. Prove that

P Q + R S = P S + Q R

Q. Assertion :Let the vectors

¯ ¯¯¯¯¯¯ ¯ P Q, ¯ ¯¯¯¯¯¯¯ ¯ Q R, ¯ ¯¯¯¯¯¯ ¯ R S, ¯ ¯¯¯¯¯ ¯ S T, ¯ ¯¯¯¯¯¯ ¯ T U

¯ ¯¯¯¯¯¯ ¯ U P

represent the sides of a regular hexagon.
STATEMENT-

1

¯ ¯¯¯¯¯¯ ¯ P Q \times (¯ ¯¯¯¯¯¯ ¯ R S + ¯ ¯¯¯¯¯ ¯ S T) \neq \to 0

. Reason: STATEMENT-

2

¯ ¯¯¯¯¯¯ ¯ P Q \times ¯ ¯¯¯¯¯¯ ¯ R S = \to 0

\to P Q \times \to S T \neq \to 0

.

Q. Six identical uniform rods

P Q, Q R, R S, S T, T U

U P

each weighing W are freely joined at their ends to form a hexagon. The rod

P Q

is fixed in a horizontal position and middle points of

P Q

S T

are connected by a vertical string. The tension in string is

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