To determine if the proportion [tex]\(\frac{e}{d} = \frac{d}{f}\)[/tex] is true, we need to analyze the given ratio and see if it holds mathematically.
1. Start with the proportion:
[tex]\[
\frac{e}{d} = \frac{d}{f}
\][/tex]
2. Cross multiply to verify equality:
[tex]\[
e \cdot f = d \cdot d
\][/tex]
3. Simplify the equation:
[tex]\[
e \cdot f = d^2
\][/tex]
The given proportion [tex]\(\frac{e}{d} = \frac{d}{f}\)[/tex] implies that the product of [tex]\(e\)[/tex] and [tex]\(f\)[/tex] should equal the square of [tex]\(d\)[/tex]. Generally, this type of relationship does not hold for arbitrary values of [tex]\(e\)[/tex], [tex]\(d\)[/tex], and [tex]\(f\)[/tex]. Therefore, the proportion [tex]\(\frac{e}{d} = \frac{d}{f}\)[/tex] is not true.
To correct the error, consider the usual property of proportions where:
[tex]\[
\frac{a}{b} = \frac{c}{d} \implies a \cdot d = b \cdot c
\][/tex]
Therefore, if we have [tex]\(\frac{e}{d}\)[/tex], it should match with another proportion of the same form. Correcting the proportion, let’s assume [tex]\(\frac{e}{d} = \frac{d}{g}\)[/tex]:
[tex]\[
\frac{e}{d} = \frac{d}{g}
\][/tex]
Cross multiplication for this proportion results in:
[tex]\[
e \cdot g = d \cdot d
\][/tex]
[tex]\[
e \cdot g = d^2
\][/tex]
This correction aligns correctly with the mathematical framework of proportions.
Given the multiple-choice options, the correct answer is:
C. false, it should be [tex]\(\frac{e}{d} = \frac{d}{g}\)[/tex]
Thus, the corrected version of the original proportion is [tex]\(\frac{e}{d} = \frac{d}{g}\)[/tex]. The correct answer is false, it should be [tex]\(\frac{e}{d} = \frac{d}{g}\)[/tex].
