Question

which of these triangles are reflections of \\(\delta abc\\)?

choose the correct answer below.

a. \\(\delta def\\) and \\(\delta ghj\\) are reflections of \\(\delta abc\\).
b. \\(\delta ghj\\) is a reflection of \\(\delta abc\\).
c. \\(\delta def\\), \\(\delta ghj\\), and \\(\delta kmn\\) are reflections of \\(\delta abc\\).
d. \\(\delta def\\) is a reflection of \\(\delta abc\\).
e. \\(\delta kmn\\) is a reflection of \\(\delta abc\\).

Explanation:

Identify coordinates of the original triangle

Using the Geometric Transformations knowledge point

$$ A = (-1, 5),\quad B = (-1, 2),\quad C = (-5, 1) $$

Analyze the transformation of triangle DEF

Using the Rigid Motion knowledge point

$$ D = (5, 5),\quad E = (5, 2),\quad F = (1, 1) $$

Comparing these coordinates to \(A, B, C\):

$$ (x, y) \to (-x, y) $$

This represents a reflection across the \(y\)-axis.

Analyze the transformation of triangle GHJ

Using the Rigid Motion knowledge point

$$ G = (-1, -5),\quad H = (-1, -2),\quad J = (-5, -1) $$

Comparing these coordinates to \(A, B, C\):

$$ (x, y) \to (x, -y) $$

This represents a reflection across the \(x\)-axis.

Analyze the transformation of triangle KMN

Using the Rigid Motion knowledge point

$$ K = (1, -5),\quad M = (1, -2),\quad N = (5, -1) $$

Comparing these coordinates to \(A, B, C\):

$$ (x, y) \to (-x, -y) $$

This represents a rotation of \(180^\circ\) about the origin, which is not a single reflection.

Determine the correct option

Using the Geometric Transformations knowledge point

$$ \Delta DEF \text{ and } \Delta GHJ \text{ are reflections of } \Delta ABC. $$

This corresponds to option A.

Answer:

• A. \(\Delta DEF\) and \(\Delta GHJ\) are reflections of \(\Delta ABC\). (Correct answer)
• B. \(\Delta GHJ\) is a reflection of \(\Delta ABC\).
• C. \(\Delta DEF\), \(\Delta GHJ\), and \(\Delta KMN\) are reflections of \(\Delta ABC\).
• D. \(\Delta DEF\) is a reflection of \(\Delta ABC\).
• E. \(\Delta KMN\) is a reflection of \(\Delta ABC\).