Question

Enter the proportional segment lengths into the boxes to verify that ¯¯¯QS¯∥MN¯ . ___ /1.5= ___ / ___?

Answer 1

Two triangles RMN & RQS are similar.
Therefore, QS // MN.

Explanation:

To prove QS is parallel to MN.

`9/1.5 = 12/2 = 6/1`
`:. (RM) / (MQ) = (RN) / (NS) color (white)(aaa) Eqn (1)`

Add 1 to both sides,
`( (RM) + (MQ) ) / ( MQ) = ((RN) + (NS))/ (NS)`
`(RQ) / (MQ) = (RS) / ( NS) color (white)(aaa) Eqn (2)`

Dividing Eqn (1) by (2),
`(RM) / ( RQ) = (RN) / ( RS)`
Two sides of triangles RMN & RQS are in the same proportion.
Therefore third sides are also in the same proportion.
`:. (MN) / ( QS) = (RM) / ( RQ) = (RN) / ( RS)`

That means both the triangles RMN & RQS are similar and hence
MN // QS.