Which sequence of transformations would yield a triangle similar to the original triangle but not congruent?
A) rotation and dilation
B) reflection and rotation
C ) translation and rotation
D) reflection and translation
all these are rigid motions, except dilation
The correct answer is option A) rotation and dilation.
To create a triangle that is similar to the original triangle (meaning the corresponding angles are equal), but not congruent (meaning the corresponding sides are not equal in length), you can use a combination of rotation and dilation.
Rotation will preserve the shape of the triangle, but the triangle will not be congruent since the corresponding sides are not equal. Dilation will change the size of the triangle, but the corresponding angles will remain the same.
Therefore, by performing a rotation and dilation on the original triangle, you can obtain a triangle that is similar to the original but not congruent.
To determine the sequence of transformations that would yield a triangle similar to the original triangle but not congruent, we need to understand the definitions of the different transformations.
1. Rotation: This transformation involves turning an object around a fixed point. The angle of rotation determines how much the object is turned.
2. Dilation: This transformation involves changing the size of an object without altering its shape. It can either enlarge or shrink the object by using a scale factor.
3. Reflection: This transformation involves flipping an object over a line, known as the line of reflection. It creates a mirror image of the original object.
4. Translation: This transformation involves moving an object without changing its shape or orientation. It involves sliding an object in any direction with a fixed distance.
Now, let's analyze each option:
A) Rotation and Dilation: This would change both the size and orientation of the triangle. Since dilation alters the triangle's size, it would not result in similarity. Therefore, this option is ruled out.
B) Reflection and Rotation: This combination of transformations would make the triangle reflect over a line and then rotate it. Reflection preserves angle measures but reverses orientations, while rotation changes the orientation. Hence, this option would yield a similar triangle but not a congruent one, as the orientation would differ.
C) Translation and Rotation: This combination involves translating the triangle in a particular direction and then rotating it around a fixed point. As translation only changes the position, not the shape or orientation, the triangle would remain similar. Therefore, this option satisfies the criteria and could yield a similar but not congruent triangle.
D) Reflection and Translation: This combination of transformations would reflect the triangle and then move it in a given direction. Reflection reverses orientation, while translation only changes the position. Therefore, this sequence could result in a congruent triangle rather than a similar one.
In conclusion, the correct sequence of transformations that would yield a triangle similar to the original triangle but not congruent is Option C: Translation and Rotation.