Question

question 5 of 5

select the correct answer.

which statement justifies that \\(\triangle abc \cong \triangle def\\)?

\\(\triangle abc\\) is stretched vertically by a factor of 2, then reflected across the x-axis.
\\(\triangle abc\\) is dilated by a factor of 2, then reflected across the y-axis.
\\(\triangle abc\\) is reflected across the y-axis, then stretched horizontally by a factor of 2.
\\(\triangle abc\\) is reflected across the x-axis, then translated to the right.

Explanation:

Identify the coordinates of both triangles

Using the Geometric Transformations knowledge point
We find the coordinates of the vertices of \(\triangle ABC\) and \(\triangle DEF\) from the grid:

• \(A(-2, 3)\), \(B(2, 3)\), \(C(2, 6)\)
• \(D(1, -3)\), \(E(5, -3)\), \(F(5, -6)\)

Analyze the side lengths of both triangles

Using the Geometric Congruence Definition knowledge point
We calculate the lengths of the corresponding sides to check for congruence:

• \(AB = 2 - (-2) = 4\) units, and \(DE = 5 - 1 = 4\) units.
• \(BC = 6 - 3 = 3\) units, and \(EF = -3 - (-6) = 3\) units.
• Since the corresponding side lengths are equal, the triangles are congruent (\(\triangle ABC \cong \triangle DEF\)).

Evaluate transformations preserving congruence

Using the Rigid Transformations knowledge point

• Congruence is preserved only by rigid transformations (translations, reflections, and rotations).
• Any transformation involving stretching or dilation changes the size of the figure, which does not preserve congruence.
• Therefore, options involving stretching or dilating are incorrect.

Determine the specific sequence of rigid transformations

Using the Reflections in the Coordinate Plane and Translation of Figures knowledge points

• Reflecting \(\triangle ABC\) across the \(x\)-axis:

\((x, y)
ightarrow (x, -y)\)
\(A(-2, 3)
ightarrow A'(-2, -3)\)
\(B(2, 3)
ightarrow B'(2, -3)\)
\(C(2, 6)
ightarrow C'(2, -6)\)

• Translating the reflected triangle \(\triangle A'B'C'\) to the right by 3 units:

\((x, -y)
ightarrow (x + 3, -y)\)
\(A'(-2, -3)
ightarrow D(1, -3)\)
\(B'(2, -3)
ightarrow E(5, -3)\)
\(C'(2, -6)
ightarrow F(5, -6)\)

• This sequence perfectly maps \(\triangle ABC\) onto \(\triangle DEF\).

Answer:

• \(\triangle ABC\) is stretched vertically by a factor of 2, then reflected across the x-axis.
• \(\triangle ABC\) is dilated by a factor of 2, then reflected across the y-axis.
• \(\triangle ABC\) is reflected across the y-axis, then stretched horizontally by a factor of 2.
• \(\triangle ABC\) is reflected across the x-axis, then translated to the right. (Correct answer)