Question
What transformations were made to triangle ABC to produce triangle XYZ? Assess whether the two triangles are congruent. How do you know? Show your work.
Triangle ABC:
Point A: (-5, 2)
Point B: (-3, 2)
Point C: (-3, 5)
Triangle XYZ:
Point X: (0, -6)
Point Y: (0, -4)
Point Z: (-3, -4)
Triangle ABC:
Point A: (-5, 2)
Point B: (-3, 2)
Point C: (-3, 5)
Triangle XYZ:
Point X: (0, -6)
Point Y: (0, -4)
Point Z: (-3, -4)
Answers
GPT-4o mini
To determine the transformations made to triangle ABC to produce triangle XYZ, we will first analyze the coordinates of both triangles.
**Coordinates of Triangle ABC:**
- Point A: (-5, 2)
- Point B: (-3, 2)
- Point C: (-3, 5)
**Coordinates of Triangle XYZ:**
- Point X: (0, -6)
- Point Y: (0, -4)
- Point Z: (-3, -4)
### Step 1: Find the transformations
**1. **Translation:**
To transform triangle ABC to triangle XYZ, we can start by examining if a translation has occurred.
Let's determine the translation vector by comparing the coordinates of corresponding points:
- From A (-5, 2) to X (0, -6):
- The change in x is \( 0 - (-5) = 5\)
- The change in y is \( -6 - 2 = -8\)
Thus, the translation vector is (5, -8).
Let's check if this translation works for the other points:
- From B (-3, 2) to Y (0, -4):
- The change in x is \( 0 - (-3) = 3\)
- The change in y is \( -4 - 2 = -6\)
**This indicates that we have an inconsistency.**
- From C (-3, 5) to Z (-3, -4):
- The change in x is \( -3 - (-3) = 0\)
- The change in y is \( -4 - 5 = -9\)
### Step 1: Check the differences in coordinates:
**Find lengths of sides:**
**Length of AB**:
\[
AB = \sqrt{((-3) - (-5))^2 + (2 - 2)^2} = \sqrt{(2)^2 + (0)^2} = 2
\]
**Length of BC**:
\[
BC = \sqrt{((-3) - (-3))^2 + (5 - 2)^2} = \sqrt{(0)^2 + (3)^2} = 3
\]
**Length of CA**:
\[
CA = \sqrt{((-5) - (-3))^2 + (2 - 5)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
### Lengths of sides for triangle XYZ:
**Length of XY**:
\[
XY = \sqrt{(0 - 0)^2 + (-4 - (-6))^2} = \sqrt{0 + (2)^2} = 2
\]
**Length of YZ**:
\[
YZ = \sqrt{(-3 - 0)^2 + (-4 - (-4))^2} = \sqrt{(-3)^2 + 0} = 3
\]
**Length of ZX**:
\[
ZX = \sqrt{(-3 - 0)^2 + (-6 - (-4))^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
### Step 2: Compare side lengths
Now we have the lengths of the sides:
**Triangle ABC:**
- AB = 2
- BC = 3
- CA = \(\sqrt{13}\)
**Triangle XYZ:**
- XY = 2
- YZ = 3
- ZX = \(\sqrt{13}\)
### Conclusion: Assessment of Congruence
Since triangle ABC and triangle XYZ have all corresponding sides that are equal:
- \(AB = XY\)
- \(BC = YZ\)
- \(CA = ZX\)
This means triangle ABC is congruent to triangle XYZ by the Side-Side-Side (SSS) criterion for congruence.
### Final Answer:
**Transformations**: A translation of \( (5, -8)\) has been applied with each side length matching. **Conclusion**: Triangle ABC is congruent to triangle XYZ.
**Coordinates of Triangle ABC:**
- Point A: (-5, 2)
- Point B: (-3, 2)
- Point C: (-3, 5)
**Coordinates of Triangle XYZ:**
- Point X: (0, -6)
- Point Y: (0, -4)
- Point Z: (-3, -4)
### Step 1: Find the transformations
**1. **Translation:**
To transform triangle ABC to triangle XYZ, we can start by examining if a translation has occurred.
Let's determine the translation vector by comparing the coordinates of corresponding points:
- From A (-5, 2) to X (0, -6):
- The change in x is \( 0 - (-5) = 5\)
- The change in y is \( -6 - 2 = -8\)
Thus, the translation vector is (5, -8).
Let's check if this translation works for the other points:
- From B (-3, 2) to Y (0, -4):
- The change in x is \( 0 - (-3) = 3\)
- The change in y is \( -4 - 2 = -6\)
**This indicates that we have an inconsistency.**
- From C (-3, 5) to Z (-3, -4):
- The change in x is \( -3 - (-3) = 0\)
- The change in y is \( -4 - 5 = -9\)
### Step 1: Check the differences in coordinates:
**Find lengths of sides:**
**Length of AB**:
\[
AB = \sqrt{((-3) - (-5))^2 + (2 - 2)^2} = \sqrt{(2)^2 + (0)^2} = 2
\]
**Length of BC**:
\[
BC = \sqrt{((-3) - (-3))^2 + (5 - 2)^2} = \sqrt{(0)^2 + (3)^2} = 3
\]
**Length of CA**:
\[
CA = \sqrt{((-5) - (-3))^2 + (2 - 5)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
### Lengths of sides for triangle XYZ:
**Length of XY**:
\[
XY = \sqrt{(0 - 0)^2 + (-4 - (-6))^2} = \sqrt{0 + (2)^2} = 2
\]
**Length of YZ**:
\[
YZ = \sqrt{(-3 - 0)^2 + (-4 - (-4))^2} = \sqrt{(-3)^2 + 0} = 3
\]
**Length of ZX**:
\[
ZX = \sqrt{(-3 - 0)^2 + (-6 - (-4))^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
### Step 2: Compare side lengths
Now we have the lengths of the sides:
**Triangle ABC:**
- AB = 2
- BC = 3
- CA = \(\sqrt{13}\)
**Triangle XYZ:**
- XY = 2
- YZ = 3
- ZX = \(\sqrt{13}\)
### Conclusion: Assessment of Congruence
Since triangle ABC and triangle XYZ have all corresponding sides that are equal:
- \(AB = XY\)
- \(BC = YZ\)
- \(CA = ZX\)
This means triangle ABC is congruent to triangle XYZ by the Side-Side-Side (SSS) criterion for congruence.
### Final Answer:
**Transformations**: A translation of \( (5, -8)\) has been applied with each side length matching. **Conclusion**: Triangle ABC is congruent to triangle XYZ.