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

\[\begin{align}  & \angle AOB=\angle COD \\  & \text{(vertically}\,\text{opposite}\,\,\text{angles)} \\  & \angle BAO=\angle DCO \\  & (\text{alternate interior angles }) \\  & \Rightarrow \Delta AOB\tilde{\ }\Delta COD \\  & (\text{ AA criterion }) \\ \end{align}\]

Hence \({ \frac{O A}{O C}=\frac{O B}{O D}}\)