Ex.6.3 Q3 Triangles Solution - NCERT Maths Class 10

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Question

Diagonals \(AC\) and \(BD\) of a trapezium \(ABCD\) with \( AB\, ||\, DC\) intersect each other at the point \(O.\) Using a similarity criterion for two triangles, show that \(\begin{align}\frac{OA}{OC}=\frac{OB}{OD}\end{align}\).

Diagram

Text Solution

Reasoning:

If two angles of one triangle are respectively equal to the two angles of another triangle, then the two triangles are similar.

This is referred to as the \(AA\) criterion.

Steps:

In \(\Delta  \rm{A O B}, \Delta \rm{C O D}\)

\[\begin{align}& {\angle A O B}={\angle C O D\,\,\, \text { (vertically opposite angles) }} \\ &{\angle B A O=\angle D C O\,\,\,( \text { alternate interior angles })} \\ &{\Rightarrow \Delta A O B \sim \Delta C O D\,\,\,(\text { AA criterion })} \\ \text{Hence,}\,\, { \frac{O A}{O C}=\frac{O B}{O D}}\end{align}\]