English, asked by abcdef12315, 10 months ago

Hola Mates ! ❤️

Question :
ABCD is a trapezium in which ABIIDC and it's diagonals intersect each other at point 'O'.
Show that AO/BO = CO/DO

Answers

Answered by Anonymous
70

\huge\mathfrak{Answer:}

Given:

  • We have been given a trapezium ABCD such that AB || DC.
  • The diagonals AC and BD intersect each other at O.

Construction:

  • Let us draw OE parallel to AB or DC.

Solution:

We have been given a trapezium ABCD such that AB || DC. The diagonals AC and BD intersect each other at O.

We have constructed OE || DC.

Now,

OE || DC [By construction]

∴ Using the Basic Proportionality Theorem, we have

 \sf{ \dfrac{AE}{ED} =  \dfrac{AO}{CO} }______________(1)

Now in △ABD,

OE || AB [ By Construction]

∴ Using the Basic Proportionality Theorem, we have

 \sf{ \dfrac{ED}{AE}  =  \dfrac{DO}{BO} }

 \implies\sf{ \dfrac{AE}{ED}  =  \dfrac{BO}{DO} }________(2)

Now, from equation (1) and (2) we get,

 \sf{ \dfrac{AE}{ED}  =  \dfrac{BO}{DO}  =  \dfrac{AO}{CO} }

 \implies\sf{ \dfrac{BO}{DO}  =  \dfrac{AO}{CO} }

\implies\sf{\dfrac{AO}{BO} = \dfrac{CO}{DO}}

Attachments:

RvChaudharY50: Perfect. ❤️
Answered by Anonymous
53

\huge\mathfrak\blue{Answer:}

Given:

  • We have been given a Trapezium ABCD in which AB II CD
  • The Diagonals AC and BD intersect at point O

To prove :

  • \dfrac{AO}{BO}=\dfrac{CO}{DO}

Construction:

  • Let us draw a line segment OX Parallel to AB or CD
  • OX II AB or OX II CD ---------( 1 )

Solution:

Since it is given that the Diagonals AC and BD of Trapezium ABCD intersect at point O.

Now , In ∆ ABC

OX II AB [ Equation 1 ]

Using Thales Theorem , we get

\dfrac{AO}{CO}=\dfrac{BX}{XC} ------------------ ( 2 )

________________________________

Now In ∆ BCD

OX II CD [ Equation 1 ]

Using Thales Theorm , we get

\dfrac{BX}{XC}=\dfrac{BO}{DO} ----------------- ( 3 )

________________________________

From Equation ( 2 ) and ( 3 ) we get

=> \dfrac{AO}{CO}=\dfrac{BO}{DO}

=>\dfrac{AO}{BO}=\dfrac{CO}{DO}

Hence Proved !!

Attachments:

RvChaudharY50: Awesome. ❤️
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