Get the most out of your questions with the extensive resources available on IDNLearn.com. Discover reliable and timely information on any topic from our network of knowledgeable professionals.
Sagot :
Understanding the transformations applied to [tex]\( \Delta XYZ \)[/tex] will help us evaluate the properties of [tex]\( \Delta XYZ \)[/tex] and [tex]\( \Delta X'Y'Z' \)[/tex].
1. Reflection over a vertical line:
- This transformation creates a mirror image of [tex]\( \Delta XYZ \)[/tex]. All angles and side lengths remain unchanged, but the orientation can reverse, resulting in a congruent but opposite triangle.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation reduces the size of [tex]\( \Delta XYZ \)[/tex] such that all side lengths are halved, but the angles remain the same. This results in a similar triangle, [tex]\( \Delta X'Y'Z' \)[/tex], with proportional sides and identical angles to [tex]\( \Delta XYZ \)[/tex].
Given the above transformations, let's analyze each statement:
1. [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]:
- True. A dilation with a scale factor retains the similarity of the triangles. The angles are congruent, and the sides are in proportion (in this case, the sides of [tex]\( \Delta X'Y'Z' \)[/tex] are [tex]\(\frac{1}{2}\)[/tex] the length of the corresponding sides of [tex]\( \Delta XYZ \)[/tex]).
2. [tex]\( \angle XZY \cong \angle YY'X' \)[/tex]:
- This statement appears incorrect because [tex]\( \angle XZY \)[/tex] (an angle in the original triangle) cannot be directly compared to [tex]\( \angle YY'X' \)[/tex], a combination of different vertices not defined in our transformed triangle.
3. [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]:
- This statement is somewhat redundant as it states that a segment is congruent to itself, which is always true, but not informative in the context of changes from the transformations. This is generally a true statement but doesn't provide relevant comparison between two different parts of our transformation process.
4. [tex]\( XZ = 2 X'Z' \)[/tex]:
- True. Since the dilation reduces all sides to [tex]\(\frac{1}{2}\)[/tex], reversing this, the original side lengths would be double those of the dilated triangle. So, [tex]\( XZ\)[/tex] from [tex]\( \Delta XYZ \)[/tex] is indeed [tex]\( 2 \times X'Z' \)[/tex] from [tex]\(\Delta X'Y'Z' \)[/tex].
5. [tex]\( m\angle YXZ = 2 m\angle YXXX \)[/tex]:
- This statement is incorrect. The dilation changes the side lengths but does not alter the angles. Therefore, the measure of angle [tex]\( YXZ \)[/tex] in [tex]\( \Delta XYZ \)[/tex] is equal to the measure of its corresponding angle in [tex]\( \Delta X'Y'Z' \)[/tex].
Thus, the correct options must be:
- [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]
- [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]
- [tex]\( XZ = 2 X'Z' \)[/tex]
1. Reflection over a vertical line:
- This transformation creates a mirror image of [tex]\( \Delta XYZ \)[/tex]. All angles and side lengths remain unchanged, but the orientation can reverse, resulting in a congruent but opposite triangle.
2. Dilation by a scale factor of [tex]\(\frac{1}{2}\)[/tex]:
- Dilation reduces the size of [tex]\( \Delta XYZ \)[/tex] such that all side lengths are halved, but the angles remain the same. This results in a similar triangle, [tex]\( \Delta X'Y'Z' \)[/tex], with proportional sides and identical angles to [tex]\( \Delta XYZ \)[/tex].
Given the above transformations, let's analyze each statement:
1. [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]:
- True. A dilation with a scale factor retains the similarity of the triangles. The angles are congruent, and the sides are in proportion (in this case, the sides of [tex]\( \Delta X'Y'Z' \)[/tex] are [tex]\(\frac{1}{2}\)[/tex] the length of the corresponding sides of [tex]\( \Delta XYZ \)[/tex]).
2. [tex]\( \angle XZY \cong \angle YY'X' \)[/tex]:
- This statement appears incorrect because [tex]\( \angle XZY \)[/tex] (an angle in the original triangle) cannot be directly compared to [tex]\( \angle YY'X' \)[/tex], a combination of different vertices not defined in our transformed triangle.
3. [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]:
- This statement is somewhat redundant as it states that a segment is congruent to itself, which is always true, but not informative in the context of changes from the transformations. This is generally a true statement but doesn't provide relevant comparison between two different parts of our transformation process.
4. [tex]\( XZ = 2 X'Z' \)[/tex]:
- True. Since the dilation reduces all sides to [tex]\(\frac{1}{2}\)[/tex], reversing this, the original side lengths would be double those of the dilated triangle. So, [tex]\( XZ\)[/tex] from [tex]\( \Delta XYZ \)[/tex] is indeed [tex]\( 2 \times X'Z' \)[/tex] from [tex]\(\Delta X'Y'Z' \)[/tex].
5. [tex]\( m\angle YXZ = 2 m\angle YXXX \)[/tex]:
- This statement is incorrect. The dilation changes the side lengths but does not alter the angles. Therefore, the measure of angle [tex]\( YXZ \)[/tex] in [tex]\( \Delta XYZ \)[/tex] is equal to the measure of its corresponding angle in [tex]\( \Delta X'Y'Z' \)[/tex].
Thus, the correct options must be:
- [tex]\( \Delta XYZ \sim \Delta X'Y'Z' \)[/tex]
- [tex]\( \overline{YX} \cong \overline{YX} \)[/tex]
- [tex]\( XZ = 2 X'Z' \)[/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for answers ends at IDNLearn.com. Thank you for visiting, and we hope to assist you again soon.
