Question

use the image to answer the question.
bella wants to use rigid transformations to show that ( \triangle ghi cong \triangle lkj ) to illustrate the sss triangle congruence criterion. which of the following could she do first?
(1 point)
translate ( \triangle ghi ) along a vector that takes point ( g ) to point ( j ).
translate ( \triangle ghi ) along a vector that takes point ( i ) to point ( k ).
translate ( \triangle ghi ) along a vector that takes point ( g ) to point ( l ).
translate ( \triangle ghi ) along a vector that takes point ( i ) to point ( l )

Explanation:

Step1: Analyze the correspondence of vertices

For \(\triangle GHI\) and \(\triangle LKJ\) to be congruent by SSS (side - side - side), we need to match the vertices correctly. In a rigid transformation (translation in this case), we want to map one triangle to the other such that the corresponding sides are equal.
We know that in \(\triangle GHI\) and \(\triangle LKJ\), we need to have \(GH = LK\), \(HI=KJ\), and \(GI = LJ\).
If we translate \(\triangle GHI\) along a vector that takes point \(I\) to point \(K\):
Let's consider the translation vector \(\overrightarrow{IK}\). A translation of \(\triangle GHI\) by \(\overrightarrow{IK}\) will move each point of \(\triangle GHI\) in the same direction and by the same distance.
If we translate point \(I\) to \(K\), then:

• The side \(HI\) (which is congruent to \(KJ\)) will be in a better position for showing the SSS congruence. After translating \(I\) to \(K\), we can then check the alignment of the other sides.

Step2: Check other options

• If we translate \(\triangle GHI\) along a vector that takes point \(G\) to point \(J\):

The correspondence of sides and vertices for SSS congruence (\(\triangle GHI\cong\triangle LKJ\)) will not be straightforward. The side - side - side relationship will not be easily visualized as the key side \(HI\) (which should correspond to \(KJ\)) will not be in a proper aligned position.

• If we translate \(\triangle GHI\) along a vector that takes point \(G\) to point \(L\):

This translation will not help in aligning the sides \(HI\) and \(KJ\) (which are congruent) and other corresponding sides (\(GI\) and \(LJ\), \(GH\) and \(LK\)) for showing SSS congruence in an intuitive way.

• If we translate \(\triangle GHI\) along a vector that takes point \(I\) to point \(L\):

The side \(HI\) (which should correspond to \(KJ\)) will not be in a position to show the SSS congruence as \(KJ\) has a different length relationship with the translated side.

Answer:

Translate \(\triangle GHI\) along a vector that takes point \(I\) to point \(K\).