Question

if PQ||MN then show that OP/ON=OQ/OM​

Answer 1

Given :-

• In quadrilateral PQNM, PQ || MN

To Prove :-

$\rm :\longmapsto\:\dfrac{OP}{ON} = \dfrac{OQ}{OM}$

Concept Used :-

• AA Similarity of triangles

$\large\underline{\bf{Solution-}}$

$\red{\rm :\longmapsto\: \bf \: In \: \triangle \: ONM \: and \: \triangle \: OPQ}$

$\rm :\longmapsto\:\angle \: OPQ=\angle \: ONM \: \{Alternate \: interior \: angles \}$

$\rm :\longmapsto\:\angle \: POQ=\angle \: NOM \: \{vertically \: opposite \: angles \}$

$\red{\bf\implies \:\triangle \: ONM \: \sim \: \triangle \: OPQ \: \: \{AA \: Similarity \}}$

$\rm :\longmapsto\:\dfrac{OP}{ON} = \dfrac{OQ}{OM} \: \: \{CPST \}$

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

$\purple{ \large{\boxed{ \bf \: Additional \: Information}}}$

1. Pythagoras Theorem :-

• This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

• This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

• This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem,

• If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.