Chapter 8, Problem 23RP

To determine

To prove:The diagonals of trapezoid divide each other proportionally.

Explanation of Solution

Given information:

A trapezium ABCD in which  AB ∥ DC  and its diagonals AC and BD intersect at O.

Formula used:

Basic proportionate theorem

Proof:

Through O draw $\text{EO}\parallel \text{AB}$ meeting at E

  

In $\Delta \text{ADC}$ , $\text{EO}\parallel \text{DC}$ ,we have,

  $\frac{\text{ED}}{\text{AE}}=\frac{\text{OC}}{\text{AO}}$ ……..(a)

In $\Delta \text{DAB}$ , $\text{EO}\parallel \text{AB}$ ,we have,

  $\frac{\text{ED}}{\text{AE}}=\frac{\text{OD}}{\text{BO}}$ ……..(b)

From (a) and (b) we have,

  $\frac{\text{OC}}{\text{AO}}=$

  $\frac{\text{OD}}{\text{BO}}$

Hence proved