What transformations were made to △ABC to produce △XYZ ? Assess whether the two triangles are congruent. How do you know?

Question

Points are:

A (2,-5)
B (2,-3)
C (5,-3)

X (0,-6)
Y (0,-4)
Z (-3,-4)

Answer 1

To determine the transformations made to triangle \( \triangle ABC \) to produce triangle \( \triangle XYZ \) and to assess the congruence of the two triangles, we can follow these steps:

1. Find the Vertices of Each Triangle:

• Triangle \( \triangle ABC \) has vertices:

  • \( A(2, -5) \)
  • \( B(2, -3) \)
  • \( C(5, -3) \)

• Triangle \( \triangle XYZ \) has vertices:

  • \( X(0, -6) \)
  • \( Y(0, -4) \)
  • \( Z(-3, -4) \)

2. Calculate Distances Between Points:

We will calculate the lengths of the sides of both triangles to compare them to see if they are congruent.

For \( \triangle ABC \):

• Length of \( AB \): \[ AB = \sqrt{(2-2)^2 + (-5 + 3)^2} = \sqrt{0 + 4} = 2 \]
• Length of \( BC \): \[ BC = \sqrt{(2-5)^2 + (-3 + 3)^2} = \sqrt{(-3)^2 + 0} = 3 \]
• Length of \( CA \): \[ CA = \sqrt{(2-5)^2 + (-5 + 3)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

For \( \triangle XYZ \):

• Length of \( XY \): \[ XY = \sqrt{(0-0)^2 + (-6 + 4)^2} = \sqrt{0 + 4} = 2 \]
• Length of \( YZ \): \[ YZ = \sqrt{(0 + 3)^2 + (-4 + 4)^2} = \sqrt{3^2 + 0} = 3 \]
• Length of \( ZX \): \[ ZX = \sqrt{(0 + 3)^2 + (-6 + 4)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

3. Compare Side Lengths:

Now let's summarize the side lengths:

• For \( \triangle ABC \):

  • \( AB = 2 \)
  • \( BC = 3 \)
  • \( CA = \sqrt{13} \)

• For \( \triangle XYZ \):

  • \( XY = 2 \)
  • \( YZ = 3 \)
  • \( ZX = \sqrt{13} \)

4. Assess Congruence:

Since all corresponding side lengths are equal:

• \( AB = XY \)
• \( BC = YZ \)
• \( CA = ZX \)

We can conclude that \( \triangle ABC \) is congruent to \( \triangle XYZ \) by the Side-Side-Side (SSS) congruence criterion.

5. Determine Transformations:

To find the transformations, we can observe the coordinates of the vertices:

• Translation: To translate \( A(2,-5) \) to \( X(0,-6) \), we subtract \( (2, -5) \) from \( (0, -6) \): \[ \text{Translation vector} = (0 - 2, -6 + 5) = (-2, -1) \]

Applying the same translation to points B and C:

• \( B(2,-3) \) becomes \( (2 - 2, -3 -1) = (0,-4) \) which is the point \( Y \).
• \( C(5,-3) \) becomes \( (5 - 2, -3 - 1) = (3,-4) \), noting that \( 3 \) in the negative direction gives \( Z(-3, -4) \).

Thus, the transformation includes a translation by vector \((-2, -1)\).

Conclusion:

In conclusion, \( \triangle ABC \) is congruent to \( \triangle XYZ \) through a transformation that consists of a translation by the vector \((-2, -1)\).