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Question
Identify the transformation happened to get △DEF from △ABC and the correct congruence relation between the triangles.
Identify the transformation happened to get △DEF from △ABC and the correct congruence relation between the triangles.
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Solution
The correct option is D Reflection along the y-axis, △EFD≅△BCA
Let’s first check for the correct transformation relation, and then for the correct congruence relation.
In Option A: Clockwise rotation of 90∘,△ABC≅△DEF
Since rotation by 90∘ involves turning the image, the triangle DEF doesn’t look like the rotated image of ABC, which means rotation is an incorrect transformation.
In Option C: Rightward translation by 6 units, △FED≅△BAC
Translation involves sliding and so the orientation of the image doesn't change, but here, DEF has a different orientation than ABC, which means this transformation is also incorrect.
In option B: Reflection along the y-axis, △DEF≅△ACB
The above figure shows a reflection along the y-axis that turns △ABC into △DEF.
Therefore, the measure of the corresponding sides and angles will be the same.
⇒AB=DE,BC=EF,CA=FD,∠A=∠D,∠B=∠E,and ∠C=∠F
The congruence relation △DEF≅△ACB implies that DE=AC, which is incorrect as we found that AB=DE.
⇒ The congruence relation △DEF≅△ACB is incorrect.
So, in option B, the transformation is correct, but the congruence is not getting satisfied. Thus, this option is also incorrect.
In option D: Reflection along the y-axis, △EFD≅△BCA
We know that △ABC is reflected along the y-axis to produce the image △DEF. (already found in option B)
According to the relation △EFD≅△BCA, we check if the corresponding sides are congruent.
Here, EF=BC,FD=CA,ED=BA, which is correct from the equalities we have in option B.
⇒△EFD≅△BCA
Here, the transformation and the congruence relation is correct. Thus, this option is correct.
➡Option D is correct.
Let’s first check for the correct transformation relation, and then for the correct congruence relation.
In Option A: Clockwise rotation of 90∘,△ABC≅△DEF
Since rotation by 90∘ involves turning the image, the triangle DEF doesn’t look like the rotated image of ABC, which means rotation is an incorrect transformation.
In Option C: Rightward translation by 6 units, △FED≅△BAC
Translation involves sliding and so the orientation of the image doesn't change, but here, DEF has a different orientation than ABC, which means this transformation is also incorrect.
In option B: Reflection along the y-axis, △DEF≅△ACB
The above figure shows a reflection along the y-axis that turns △ABC into △DEF.
Therefore, the measure of the corresponding sides and angles will be the same.
⇒AB=DE,BC=EF,CA=FD,∠A=∠D,∠B=∠E,and ∠C=∠F
The congruence relation △DEF≅△ACB implies that DE=AC, which is incorrect as we found that AB=DE.
⇒ The congruence relation △DEF≅△ACB is incorrect.
So, in option B, the transformation is correct, but the congruence is not getting satisfied. Thus, this option is also incorrect.
In option D: Reflection along the y-axis, △EFD≅△BCA
We know that △ABC is reflected along the y-axis to produce the image △DEF. (already found in option B)
According to the relation △EFD≅△BCA, we check if the corresponding sides are congruent.
Here, EF=BC,FD=CA,ED=BA, which is correct from the equalities we have in option B.
⇒△EFD≅△BCA
Here, the transformation and the congruence relation is correct. Thus, this option is correct.
➡Option D is correct.
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