Triangle BAC was rotated 90° clockwise and dilated at a scale factor of 2 from the origin to create triangle XYZ. Based on these transformations, which statement is true?

Question

coordinate plane with triangles ACB and XYZ with A at negative 4 comma 4, C at negative 1 comma 3, B at negative 4 comma 0, X at 0 comma 8, Y at 8, comma 8, and Z at 6 comma 2

∠B ≅ ∠X
∠A ≅ ∠Z
∠C ≅ ∠X
∠Y ≅ ∠B

Answer 1

To determine which statement is true based on the transformations of triangle BAC to triangle XYZ, we'll examine the angles closely.

First, let's note the given coordinates for triangle BAC:

• A: (-4, 4)
• B: (-4, 0)
• C: (-1, 3)

Now, let's find the angles of triangle BAC before the transformations.

Next, we will consider the transformations:

1. Rotation of 90° clockwise around the origin:

   • This transformation takes a point (x, y) to (y, -x).
   • Applying this to point A (-4, 4):
     • New point X = (4, 4)
   • Applying this to point B (-4, 0):
     • New point Y = (0, 4)
   • Applying this to point C (-1, 3):
     • New point Z = (3, 1)

2. Dilation at a scale factor of 2:

   • This transformation scales all coordinates by 2.
   • Applying this scaling to point X (4, 4):
     • New point = (8, 8)
   • Applying this scaling to point Y (0, 4):
     • New point = (0, 8)
   • Applying this scaling to point Z (3, 1):
     • New point = (6, 2)

Thus, we find the vertices of triangle XYZ to be:

• X: (8, 8)
• Y: (0, 8)
• Z: (6, 2)

Now comparing the angles:

1. ** ∠B ≅ ∠X** (B corresponds to Y after the transformations)
2. ** ∠A ≅ ∠Z** (A corresponds to X after transformations)
3. ** ∠C ≅ ∠X** (C corresponds to Y after transformations)
4. ** ∠Y ≅ ∠B** (Y corresponds to B after transformations)

So to find the corresponding angles with the original triangle's positions:

• A corresponds to X
• B corresponds to Y
• C corresponds to Z

The true statement from the given options appears to be ** ∠A ≅ ∠Z** based on their corresponding points pre and post transformations.