Question

diagonal ac and bd of trapisum abcd with ab|| dc intersect each other at the point show that OA/OC=OB/OD​

Answer 1

Given :-

• A trapezium ABCD in which AB || CD and diagonals AC and BD intersects at O.

To prove :-

$\rm :\longmapsto\:\dfrac{OA}{OC} = \dfrac{OB}{OD}$

Concept Used :-

• Similarity of triangles

Proof :-

Given that,

In trapezium ABCD,

• AB || CD

• Diagonals AC and BD intersects at O.

$\red{\rm :\longmapsto\:In \: \triangle \: AOB \: and \: \triangle \: COD}$

$\rm :\longmapsto\:\angle \: AOB \: = \: \angle \: COD \: \{vertically \: opp. \: angles \}$

$\rm :\longmapsto\:\angle \: ABO \: = \: \angle \: CDO \: \{alternate \: int. \: angles \}$

$\red{{\rm :\implies\: \: \triangle \: AOB \: \sim \: \triangle \: COD} \: \: \: \{ \sf \: AA \: Similarity \}}$

$\bf\implies \:\dfrac{OA}{OC} = \dfrac{OB}{OD} \: \: \: \: \: (CPST)$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. Basic Proportionality Theorem :- It states that a line is drawn to parallel to third side, intersects the two sides of a triangle in two distinct points, the line is divided in the same ratio.

2. Pythagoras Theorem :- It states that In right angle triangle, the square of the hypotenuse is equal to sum of the squares of remaining two sides.

3. Converse of Pythagoras Theorem :- In a triangle, if square of longest side is equal to sum of the squares of remaining two sides, then angle opposite to longest side is always 90°.

4. Area Ratio Theorem :- This theorem states that ratio of area of two similar triangles is equal to ratio of the squares of their corresponding sides.