Question
Use the image to answer the question.
Two triangles, upper A upper B upper C and Upper X upper Y upper Z, are graphed on a coordinate plane. The x-axis ranges from negative 8 to 4 in increments of 1. The y-axis ranges from negative 8 to 6 in increments of 1.
What transformations were made to △ABC to produce △XYZ ? Assess whether the two triangles are congruent. How do you know? Show your work.
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Two triangles, upper A upper B upper C and Upper X upper Y upper Z, are graphed on a coordinate plane. The x-axis ranges from negative 8 to 4 in increments of 1. The y-axis ranges from negative 8 to 6 in increments of 1.
What transformations were made to △ABC to produce △XYZ ? Assess whether the two triangles are congruent. How do you know? Show your work.
(2 points)
BoldItalicUnderlineIncrease indentDecrease indentFormula keypadUndoRedo
Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.
The answer is fully supported by details.
There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.
1
The answer is somewhat clear.
Details are weak or not related.
There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.
0 The question is not answered.
Skip to navigation
page 1 of 1
Answers
GPT-4o mini
To determine the transformations applied to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \), we can analyze the coordinates of the vertices of both triangles on the coordinate plane.
Assuming:
- The vertices of \( \triangle ABC \) are \( A(a_1, b_1) \), \( B(a_2, b_2) \), and \( C(a_3, b_3) \).
- The vertices of \( \triangle XYZ \) are \( X(x_1, y_1) \), \( Y(x_2, y_2) \), and \( Z(x_3, y_3) \).
1. **Identifying Transformations**:
- **Translation**: If the coordinates of \( \triangle XYZ \) are obtained by adding a constant to the corresponding coordinates of \( \triangle ABC \), the transformation is a translation.
- **Rotation**: If the points \( \triangle XYZ \) are the result of rotating \( \triangle ABC \) around a specific point by a given angle, that is a rotation.
- **Reflection**: If \( \triangle XYZ \) is the mirror image of \( \triangle ABC \) across a certain line (like the x-axis or y-axis), then it is a reflection.
- **Dilations**: If the size of the triangles differs, a dilation may be involved.
2. **Congruence**: Two triangles are congruent if:
- They have the same shape and size.
- Their corresponding sides are equal in length and their corresponding angles are equal.
To confirm congruence, we would typically measure the length of the sides of both triangles and their angles (using the distance formula for sides and properties of angles).
**Conclusion**:
- **Transformations Applied**: If we determine through an analysis of coordinates that the vertices of \( \triangle ABC \) were moved consistently (say, shifted right and up by a certain amount), we can conclude a translation occurred. Additionally, if the lengths of the sides of both triangles are equal, then the triangles are congruent.
- **Assessment of Congruence**: If after measuring the sides (\( AB = XY \), \( BC = YZ \), and \( CA = ZX \)) we find them equal, and all angles are unchanged, then \( \triangle ABC \) is congruent to \( \triangle XYZ \).
To summarize:
- **Transformations**: [State the specific transformation(s) based on observed coordinates].
- **Congruent**: Yes or No, based on whether side lengths and angles correspond.
**Example**:
If \( A(0, 0) \to X(2, 2) \), \( B(1, 1) \to Y(3, 3) \), \( C(0, 2) \to Z(2, 4) \), there is a translation of \( (+2, +2) \). If all sides of \( \triangle ABC \) equal the sides of \( \triangle XYZ \), then they are congruent triangles.
By detailing the findings from the figures and confirming the criteria for congruency, a strong response can be provided.
Assuming:
- The vertices of \( \triangle ABC \) are \( A(a_1, b_1) \), \( B(a_2, b_2) \), and \( C(a_3, b_3) \).
- The vertices of \( \triangle XYZ \) are \( X(x_1, y_1) \), \( Y(x_2, y_2) \), and \( Z(x_3, y_3) \).
1. **Identifying Transformations**:
- **Translation**: If the coordinates of \( \triangle XYZ \) are obtained by adding a constant to the corresponding coordinates of \( \triangle ABC \), the transformation is a translation.
- **Rotation**: If the points \( \triangle XYZ \) are the result of rotating \( \triangle ABC \) around a specific point by a given angle, that is a rotation.
- **Reflection**: If \( \triangle XYZ \) is the mirror image of \( \triangle ABC \) across a certain line (like the x-axis or y-axis), then it is a reflection.
- **Dilations**: If the size of the triangles differs, a dilation may be involved.
2. **Congruence**: Two triangles are congruent if:
- They have the same shape and size.
- Their corresponding sides are equal in length and their corresponding angles are equal.
To confirm congruence, we would typically measure the length of the sides of both triangles and their angles (using the distance formula for sides and properties of angles).
**Conclusion**:
- **Transformations Applied**: If we determine through an analysis of coordinates that the vertices of \( \triangle ABC \) were moved consistently (say, shifted right and up by a certain amount), we can conclude a translation occurred. Additionally, if the lengths of the sides of both triangles are equal, then the triangles are congruent.
- **Assessment of Congruence**: If after measuring the sides (\( AB = XY \), \( BC = YZ \), and \( CA = ZX \)) we find them equal, and all angles are unchanged, then \( \triangle ABC \) is congruent to \( \triangle XYZ \).
To summarize:
- **Transformations**: [State the specific transformation(s) based on observed coordinates].
- **Congruent**: Yes or No, based on whether side lengths and angles correspond.
**Example**:
If \( A(0, 0) \to X(2, 2) \), \( B(1, 1) \to Y(3, 3) \), \( C(0, 2) \to Z(2, 4) \), there is a translation of \( (+2, +2) \). If all sides of \( \triangle ABC \) equal the sides of \( \triangle XYZ \), then they are congruent triangles.
By detailing the findings from the figures and confirming the criteria for congruency, a strong response can be provided.