Problem 45 - Algebra 2

Question

REASONING Determine whether the following statement is sometimes, always, or never true. Explain your reasoning. The image of a dilation is congruent to its preimage.

Step-by-Step Solution

Answer

The statement is sometimes true, specifically when the scale factor is 1.

Step 1: Understanding the Problem

First, we need to understand what dilation and congruence mean. Dilation is a transformation that changes the size of a figure but not its shape. Congruence means that figures have the same shape and size.

Step 2: Analyzing Dilation

In dilation, the image is obtained by enlarging or reducing the preimage by a scale factor. The scale factor determines how much larger or smaller the image will be compared to the preimage.

Step 3: Identifying Scale Factor Condition

For the image of a dilation to be congruent to the preimage, the scale factor must be exactly 1. If the scale factor is 1, the size of the image does not change, so the image is congruent to the preimage.

Step 4: Determining Other Cases

If the scale factor is not 1, then the image will not be the same size as the preimage, and therefore, it cannot be congruent. This means congruence is not achieved with any other scale factor.

Key Concepts

Congruence

Congruence is a concept in geometry that describes two figures having the exact same size and shape. For two figures to be congruent, every angle must match and every side must be identical in length. It's like having two identical puzzle pieces; they fit and look exactly the same. When a figure is moved around, flipped, or rotated, without changing its size, it remains congruent with itself. But if the size changes, even if the shape stays the same, congruence is lost. This is important when discussing transformations like dilation because dilation can change the size of the figure.

Scale Factor

The scale factor is a crucial term in the context of dilation. It tells us how much a figure is going to be resized. If the scale factor is greater than 1, the figure grows larger. If it's between 0 and 1, the figure shrinks. If the scale factor is exactly 1, the size of the figure remains unchanged. When a figure undergoes dilation, this scale number multiplies with the original dimensions to give the new dimensions.

• A scale factor greater than 1 means the image is an enlargement.
• A scale factor less than 1 means the image is a reduction.

Recognizing these changes helps us determine whether a dilated image can be congruent with its original, or preimage. In our case, congruence is only possible if the scale factor is exactly 1.

Transformation

Transformation in geometry refers to changing a figure's position or size through various methods like translation, rotation, reflection, and dilation. Each type of transformation has a unique effect on the figure. Dilation, specifically, involves resizing a figure either up or down while retaining its proportion. It maintains the figure's shape but not necessarily its size unless the scale is 1. Here's why understanding transformation is valuable:

• It helps in visualizing how a figure changes.
• Makes it easier to derive whether the altered figure can be congruent.

Dilation, as a transformation, alters size and is directly tied to the scale factor, showing whether the resulting figure stays congruent or not with the original. For a dilation to result in congruence, only when the image's size remains unchanged, by a scale factor of 1, will the preimage and image be congruent.