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Cyclic Quadrilateral

The diagonals...

Question

The diagonals of a quadrilateral ABCD intersect each other at the point O such that $\frac{Δ A O B}{Δ B O C}$ $=$ $\frac{Δ A O D}{Δ D O C}$ then ABCD is a ......... .

Parallelogram

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Trapezium

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Square

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All of These

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Solution

The correct option is D All of These
If the trapezium is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O, then the area of AOD is equal to that of BOC, and the product of the areas of AOD and BOC is equal to that of AOB and COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides

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Similar questions

In the given, ABCD is a quadrilateral with diagonals AC and BD intersecting at point O. Hence, area of $Δ$ AOD $\times$ area of $Δ$ BOC = area of $Δ$ AOB $\times$ area of $Δ$ COD.

Q. The diagonals of a quadrilateral

A B C D

intersect each other at the point

O

, such that

\frac{A O}{B O} = \frac{C O}{D O} .

Then

A B C D

is a trapezium.

Q. Diagonals AC and BD of quadrilateral ABCD intersect at 'O', such that OB = OD. If AB = CD, then show that.

(i) Ar

(Δ D O C)

= AR

(Δ A O B)

(ii) Ar

(Δ D C B)

= Ar

(Δ A C B)

(iii) AD || BC

Q. The diagonals of a quadrilateral

A B C D

intersects each other at the point

O

such that

\frac{A O}{B O} = \frac{C O}{D O}

. Show that

A B C D

is a trapezium.

Q. Question 10
The diagonals of a quadrilateral ABCD intersect each other at the point O such that

\frac{A O}{B O} = \frac{C O}{D O}

. Show that ABCD is a trapezium.

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The diagonals of a quadrilateral ABCD intersect each other at the point O such that ΔAOBΔBOC = ΔAODΔDOC then ABCD is a ......... .

The diagonals of a quadrilateral ABCD intersect each other at the point O such that $\frac{Δ A O B}{Δ B O C}$ $=$ $\frac{Δ A O D}{Δ D O C}$ then ABCD is a ......... .