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

Answer 1

To determine whether triangles △ABC and △XYZ are congruent and to identify the transformations that took place to produce △XYZ from △ABC, we need specific information about both triangles, including their corresponding side lengths, angles, or any transformations applied (such as translation, rotation, reflection, or dilation).

Here’s a general approach to follow:

1. Identify the Transformations:

Transformations that can be performed on a triangle may include:

• Translation: Moving the triangle to a different location without changing its shape or orientation.
• Rotation: Turning the triangle around a fixed point.
• Reflection: Flipping the triangle over a line.
• Dilation: Resizing the triangle while keeping its shape.

2. Assessing Congruence:

To assess whether the two triangles are congruent, we can use one of the following criteria:

• Side-Side-Side (SSS) Congruence Postulate: If all three sides of one triangle are equal to the three sides of another triangle, then the triangles are congruent.
• Side-Angle-Side (SAS) Congruence Postulate: If two sides of one triangle and the included angle are equal to two sides of another triangle and the included angle, then the triangles are congruent.
• Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
• Angle-Angle-Side (AAS) Congruence Postulate: If two angles and a non-included side of one triangle are equal to two angles and a corresponding non-included side of another triangle, then the triangles are congruent.
• Hypotenuse-Leg (HL) Theorem: For right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another triangle, the triangles are congruent.

3. Example Work:

Assuming you have information regarding the lengths of the sides and the measures of the angles for both triangles, you would compare corresponding sides and angles directly:

1. Measure sides:

   • If \(AB = XY\), \(BC = YZ\), and \(AC = XZ\), we would check all sides.

2. Measure angles:

   • If \(∠A = ∠X\), \(∠B = ∠Y\), and \(∠C = ∠Z\), we would check all angles.

Conclusion:

1. If you can verify that either SSS, SAS, or one of the other congruence criteria holds true for the two triangles based on your observations, you can conclude that △ABC is congruent to △XYZ.

2. Detail any transformations you visually identified (like shifting or rotating) and how they relate to the figures being congruent.

Remember, without the specific dimensions and properties of triangles A, B, X, Y, Z, it's essential to apply the above principles to analyze the specific case you have. If you provide those details, I'd be happy to assist further!

Answer 2

The coordinates of ABC are, (-5,2) (-3,2) (-3,5). The coordinates of XYZ is, (0,-6) (0,-4) (-3,-4)

Answer 3

To determine the transformations that produced triangle \( \triangle ABC \) from triangle \( \triangle XYZ \) and to assess their congruence, we will analyze the coordinates provided.

Step 1: Identify Coordinates

• \( \triangle ABC \):

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

• \( \triangle XYZ \):

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

Step 2: Determine the Side Lengths

We will calculate the lengths of the sides of both triangles using the distance formula:

Distance Formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Triangle \(ABC\)

1. \( AB \): \[ AB = \sqrt{((-3) - (-5))^2 + (2 - 2)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]

2. \( BC \): \[ BC = \sqrt{((-3) - (-3))^2 + (5 - 2)^2} = \sqrt{(0)^2 + (3)^2} = \sqrt{9} = 3 \]

3. \( AC \): \[ AC = \sqrt{((-3) - (-5))^2 + (5 - 2)^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

Triangle \(XYZ\)

1. \( XY \): \[ XY = \sqrt{(0 - 0)^2 + (-4 - (-6))^2} = \sqrt{(0)^2 + (2)^2} = \sqrt{4} = 2 \]

2. \( YZ \): \[ YZ = \sqrt{((-3) - 0)^2 + (-4 - (-4))^2} = \sqrt{(-3)^2 + (0)^2} = \sqrt{9} = 3 \]

3. \( XZ \): \[ XZ = \sqrt{((-3) - 0)^2 + (-4 - (-6))^2} = \sqrt{(-3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \]

Step 3: Compare Side Lengths

Now we compare the side lengths of both triangles:

• \( \triangle ABC \):

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

• \( \triangle XYZ \):

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

From the calculations, we see:

• \( AB = XY = 2 \)
• \( BC = YZ = 3 \)
• \( AC = XZ = \sqrt{13} \)

Since all corresponding sides are equal, \( \triangle ABC \) is congruent to \( \triangle XYZ \) by the Side-Side-Side (SSS) Congruence Postulate.

Step 4: Determine Transformations

To describe the transformations from \( \triangle ABC \) to \( \triangle XYZ \):

1. Translation: To bring triangle \( ABC \) to triangle \( XYZ \), we need to translate the points.

   The translation vector can be determined by finding where each point of triangle \( ABC \) moves to triangle \( XYZ \):

   • Points \( A(-5, 2) \) to \( X(0, -6) \): Moving right \( 5 \) units and down \( 8 \) units.

   • This gives us a translation vector of \( T(5, -8) \).

   • We can confirm this movement:

     • \( A(-5, 2) + T(5, -8) = (0, -6) \)
     • \( B(-3, 2) + T(5, -8) = (2, -6) \) -- Not matching
     • However, after moving \( B \) using the same vector, we find there's likely another movement combined (maybe rotation).

In conclusion, we confirm that triangles \( ABC \) and \( XYZ \) are congruent, based on equal corresponding side lengths and conclude that a combination of translation and potentially rotation (or reflection) would result in the transformation from triangle \( ABC \) to triangle \( XYZ \).