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
△
A
B
C
to produce △XYZ
△
X
Y
Z
? Assess whether the two triangles are congruent. How do you know? Show your work.
What transformations were made to △ABC
△
A
B
C
to produce △XYZ
△
X
Y
Z
? Assess whether the two triangles are congruent. How do you know? Show your work.
Answers
Answered by
GPT-4o mini
To determine what transformations were made to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \), we will need to examine the coordinates of the vertices of both triangles. Since the specific coordinates of the vertices have not been provided, I will outline the steps you would need to follow in assessing the transformations and congruence between the two triangles.
### Steps to Identify Transformations
1. **Identify the Coordinates**: Write down the coordinates of the vertices for both triangles. For example, if \( A(a_1, b_1) \), \( B(b_1, b_2) \), and \( C(c_1, c_2) \) are coordinates for triangle \( \triangle ABC \) and \( X(x_1, y_1) \), \( Y(y_1, y_2) \), and \( Z(z_1, z_2) \) are the coordinates for triangle \( \triangle XYZ \).
2. **Compare the Coordinates**: Next, compare the coordinates of the triangles. Check for:
- **Translation**: If the triangles are in the same orientation but in different locations, then a translation is involved. You can calculate the translation vector by subtracting the coordinates of triangle \( ABC \) from those of triangle \( XYZ \).
- **Reflection/Rotation**: If there is a change in orientation (like pointing in the opposite direction), a reflection or rotation might have occurred. Analyze whether the angles between the sides remain the same and check for congruency.
- **Scaling**: If one triangle appears larger or smaller than the other, there may be a uniform scaling factor. You would calculate the distance between corresponding points in both triangles to check if they maintain the same ratio.
### Assessing Congruence
To determine if \( \triangle ABC \) and \( \triangle XYZ \) are congruent, we can use the following criteria:
1. **Side-Side-Side (SSS)**: If the lengths of all three sides of triangle \( ABC \) are equal to the lengths of the corresponding sides of triangle \( XYZ \), then the triangles are congruent.
2. **Side-Angle-Side (SAS)**: If two sides of triangle \( ABC \) are equal to two sides of triangle \( XYZ \), and the included angle is equal, then the triangles are congruent.
3. **Angle-Side-Angle (ASA)**: If two angles of triangle \( ABC \) are equal to two angles of triangle \( XYZ \), with the included side equal, the triangles are congruent.
4. **Angle-Angle-Side (AAS)**: If two angles and a non-included side of triangle \( ABC \) match those of triangle \( XYZ \), the triangles are congruent.
5. **Hypotenuse-Leg (HL)**: If both triangles are right triangles, and the hypotenuse and one leg are equal, then the triangles are congruent.
### Example Calculation (Hypothetical)
Assuming the coordinates are:
- \( A(0, 0), B(3, 0), C(0, 4) \)
- \( X(1, 1), Y(4, 1), Z(1, 5) \)
1. **Check translations**: Observe how to translate \( A \) to \( X \), \( B \) to \( Y \), and \( C \) to \( Z \).
2. **Side lengths**: Calculate lengths using the distance formula:
- Length \( AB = \sqrt{(3-0)^2 + (0-0)^2} = 3 \)
- Length \( AC = \sqrt{(0-0)^2 + (4-0)^2} = 4 \)
- Length \( BC = \sqrt{(3-0)^2 + (0-4)^2} = \sqrt{9 + 16} = 5 \)
3. Perform similar calculations for \( XYZ \).
4. **Determine if the sides match in length**; if yes, the triangles are congruent.
In conclusion, identify the transformations using the vertex coordinates of both triangles. Utilize congruence conditions to assess if the two triangles are congruent based on equal side lengths or angles. If all conditions are satisfied, the triangles are indeed congruent.
### Steps to Identify Transformations
1. **Identify the Coordinates**: Write down the coordinates of the vertices for both triangles. For example, if \( A(a_1, b_1) \), \( B(b_1, b_2) \), and \( C(c_1, c_2) \) are coordinates for triangle \( \triangle ABC \) and \( X(x_1, y_1) \), \( Y(y_1, y_2) \), and \( Z(z_1, z_2) \) are the coordinates for triangle \( \triangle XYZ \).
2. **Compare the Coordinates**: Next, compare the coordinates of the triangles. Check for:
- **Translation**: If the triangles are in the same orientation but in different locations, then a translation is involved. You can calculate the translation vector by subtracting the coordinates of triangle \( ABC \) from those of triangle \( XYZ \).
- **Reflection/Rotation**: If there is a change in orientation (like pointing in the opposite direction), a reflection or rotation might have occurred. Analyze whether the angles between the sides remain the same and check for congruency.
- **Scaling**: If one triangle appears larger or smaller than the other, there may be a uniform scaling factor. You would calculate the distance between corresponding points in both triangles to check if they maintain the same ratio.
### Assessing Congruence
To determine if \( \triangle ABC \) and \( \triangle XYZ \) are congruent, we can use the following criteria:
1. **Side-Side-Side (SSS)**: If the lengths of all three sides of triangle \( ABC \) are equal to the lengths of the corresponding sides of triangle \( XYZ \), then the triangles are congruent.
2. **Side-Angle-Side (SAS)**: If two sides of triangle \( ABC \) are equal to two sides of triangle \( XYZ \), and the included angle is equal, then the triangles are congruent.
3. **Angle-Side-Angle (ASA)**: If two angles of triangle \( ABC \) are equal to two angles of triangle \( XYZ \), with the included side equal, the triangles are congruent.
4. **Angle-Angle-Side (AAS)**: If two angles and a non-included side of triangle \( ABC \) match those of triangle \( XYZ \), the triangles are congruent.
5. **Hypotenuse-Leg (HL)**: If both triangles are right triangles, and the hypotenuse and one leg are equal, then the triangles are congruent.
### Example Calculation (Hypothetical)
Assuming the coordinates are:
- \( A(0, 0), B(3, 0), C(0, 4) \)
- \( X(1, 1), Y(4, 1), Z(1, 5) \)
1. **Check translations**: Observe how to translate \( A \) to \( X \), \( B \) to \( Y \), and \( C \) to \( Z \).
2. **Side lengths**: Calculate lengths using the distance formula:
- Length \( AB = \sqrt{(3-0)^2 + (0-0)^2} = 3 \)
- Length \( AC = \sqrt{(0-0)^2 + (4-0)^2} = 4 \)
- Length \( BC = \sqrt{(3-0)^2 + (0-4)^2} = \sqrt{9 + 16} = 5 \)
3. Perform similar calculations for \( XYZ \).
4. **Determine if the sides match in length**; if yes, the triangles are congruent.
In conclusion, identify the transformations using the vertex coordinates of both triangles. Utilize congruence conditions to assess if the two triangles are congruent based on equal side lengths or angles. If all conditions are satisfied, the triangles are indeed congruent.