- The composition of two dilations \(D_{O, 2}\) and \(D_{O, 0.75}\) is a single dilation \(D_{O, 1.5}\) with scale factor \(k=1.5\).
- A dilation with \(k=1.5\) preserves angle measures, so \(\angle M \cong \angle M''\) must be true.
- A dilation creates a similar triangle, so \(\triangle LMN \sim \triangle L''M''N''\) must be true.
- Congruence requires \(|k|=1\), which is not the case here, so \(\triangle LMN \cong \triangle L''M''N''\) is false.
- Specific coordinates for vertices cannot be determined without the initial vertex positions.
- Final answer: \(\boxed{\angle M \cong \angle M'' \text{ and } \triangle LMN \sim \triangle L''M''N''}\).
Explanation
- Analyze the composition of dilations.
To understand the transformation applied to \(\triangle LMN\), we first define the composition of the two dilations. A dilation centered at the origin, \(D_{O, k}\), maps a point \((x, y)\) to \((kx, ky)\).
We are given two dilations:
- \(D_{O, 2}(x, y) = (2x, 2y)\)
- \(D_{O, 0.75}(x, y) = (0.75x, 0.75y)\)
The composition \(D_{O, 0.75} \circ D_{O, 2}\) applied to a point \((x, y)\) is:
\(D_{O, 0.75}(D_{O, 2}(x, y)) = D_{O, 0.75}(2x, 2y) = (0.75 \times 2x, 0.75 \times 2y) = (1.5x, 1.5y)\).
This is equivalent to a single dilation \(D_{O, 1.5}\) with a scale factor of \(k = 1.5\).
- Evaluate properties of the transformation.
A dilation is a similarity transformation. This means it preserves the shape of geometric figures but changes their size if the scale factor \(k
eq 1\).
- Angle Congruence: Since a dilation preserves angle measures, the corresponding angles of the original triangle \(\triangle LMN\) and the transformed triangle \(\triangle L''M''N''\) remain congruent. Therefore, \(\angle M \cong \angle M''\) is always true.
- Similarity: By definition, a transformation that maps a figure to another figure such that corresponding angles are equal and corresponding sides are proportional is a similarity. Dilations are precisely such transformations, so \(\triangle LMN \sim \triangle L''M''N''\) is always true.
- Congruence: Two triangles are congruent if they have the same size and shape (scale factor \(k = 1\) or \(k = -1\)). Since the scale factor here is \(k = 1.5\), the size of the triangle changes. Therefore, \(\triangle LMN \cong \triangle L''M''N''\) is false.
- Evaluate coordinate-based statements.
The problem provides statements about the coordinates of vertices \(L'', M'',\) and \(N''\). However, the coordinates of the original vertices \(L, M,\) and \(N\) are not given. Without the initial coordinates, it is impossible to determine the specific locations of the transformed vertices. Therefore, the statements regarding specific coordinates of \(L'', M'',\) and \(N''\) cannot be determined to be true and must be considered false based on the provided information.
- Final conclusions.
Based on the analysis, the statements that must be true are:
- \(\angle M \cong \angle M''\)
- \(\triangle LMN \sim \triangle L''M''N''\)
The final answer is:
\(\boxed{\text{Only } \angle M \cong \angle M'' \text{ and } \triangle LMN \sim \triangle L''M''N'' \text{ must be true.}}\)
Examples
Understanding dilations and similarity is essential in fields like graphic design and architecture, where scaling objects is a daily task. For instance, if you are designing a digital logo and decide to scale it by a factor of 2, and then adjust it by a factor of 0.75, you can instantly determine that the overall change is a 1.5x increase in size. Recognizing that this transformation preserves angles and keeps the logo similar to its original form ensures that your branding remains consistent regardless of the size.