Let's go through the problem step-by-step to determine which proportion correctly represents the relationship between the dimensions of the original and the reduced picture, maintaining the same aspect ratio.
1. Original Dimensions: The width (W1) of the original picture is 10 inches, and the length (L1) is 12 inches.
2. Reduced Dimensions: The reduced width (W2) is given as 9 inches, and we need to find the reduced length (L2), which we will call [tex]\( x \)[/tex].
To find the correct proportion, we must use the fact that the aspect ratio of the picture remains the same after reduction. The aspect ratio is the ratio of width to length of the picture.
3. Setting Up the Proportions:
- The original aspect ratio is [tex]\(\frac{W1}{L1} = \frac{10}{12}\)[/tex].
- The reduced aspect ratio should be [tex]\(\frac{W2}{x} = \frac{9}{x}\)[/tex].
4. Equating the aspect ratios:
[tex]\[
\frac{10}{12} = \frac{9}{x}
\][/tex]
Looking at the options, the proportion that correctly represents this relationship is:
[tex]\[ \boxed{C} \][/tex]
Option C states [tex]\(\frac{10}{12} = \frac{x}{9}\)[/tex]. By solving this proportion, we can find the unknown length [tex]\( x \)[/tex]:
[tex]\[
\frac{10}{12} = \frac{x}{9}
\][/tex]
By cross-multiplying, we get:
[tex]\[
10 \cdot 9 = 12 \cdot x
\][/tex]
[tex]\[
90 = 12x
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{90}{12} = 7.5 \text{ inches}
\][/tex]
So, the reduced length [tex]\( x \)[/tex] is 7.5 inches. The correct proportion to use is therefore option C:
[tex]\[ \boxed{3} \][/tex]
