To determine which statement is true, we need to consider the concepts of similar figures and the relationship between their volumes.
- If two figures are similar, their volumes have a ratio that is the cube of the ratio of their corresponding linear dimensions (lengths, widths, heights, etc.).
- For example, if the ratio of the linear dimensions of two similar figures is \( k \), then the ratio of their volumes will be \( k^3 \).
In the choices given, we have two volume ratios mentioned - \( 125:1 \) and \( 5:1 \).
- The volume ratio of \( 125:1 \) suggests that the scale factor for the linear dimensions is \( \sqrt[3]{125} = 5 \).
- The volume ratio of \( 5:1 \) suggests that the scale factor for the linear dimensions is \( \sqrt[3]{5} \), which is not a whole number.
Given this information, if the volume of one figure is \( 125 \) times that of another, the two figures would be similar due to the consistent linear scale factor.
The correct conclusion, based on the information provided, would be:
The two figures are similar because the volumes have a ratio of 125:1.
You may choose between the following statements:
- "The two figures are similar because the volumes have a ratio of 125:1."
- "The two figures are similar because the volumes have a ratio of ratio of 125 to 1."
Both are essentially the same statement.
9 answers