To determine which proportional statements are true for finding the value of \(x\) in the enlargement of the triangle with given side lengths, we can apply the principle of proportionality for similar triangles.
We have the side lengths from the smaller triangle: 9 mm and 6 mm, and from the larger triangle: 16 mm and \(x\) mm. When two triangles are similar, the ratios of corresponding sides are equal.
Let’s denote the sides of the smaller triangle as \(a = 9\) mm and \(b = 6\) mm, and the sides of the larger triangle as \(c = x\) mm and \(d = 16\) mm.
The proportional relationships that can be set up between the corresponding sides are:
- \(\frac{6}{16} = \frac{9}{x}\)
- \(\frac{9}{16} = \frac{x}{6}\)
- \(\frac{9}{6} = \frac{x}{16}\)
- \(\frac{9}{x} = \frac{16}{6}\)
- \(\frac{9}{x} = \frac{6}{16}\)
Now, let's analyze each option:
- \(\frac{6}{x} = \frac{9}{16}\) - True
- \(\frac{6}{16} = \frac{9}{x}\) - True (Cross multiplying gives \(6x = 144\), so \(x = 24\) which is valid.)
- \(\frac{9}{6} = \frac{x}{16}\) - True (Cross multiplying gives \(9 \cdot 16 = 6x\), which leads to \(x = 24\).)
- \(\frac{9}{x} = \frac{16}{6}\) - False (This rearranges to \(9 \cdot 6 \neq 16 \cdot x\).)
- \(\frac{9}{x} = \frac{6}{16}\) - False (This does not hold as it suggests \(9 \cdot 16 \neq 6x\).)
Thus, the true proportional statements for finding the value of \(x\) are:
- 1: \(\frac{6}{x} = \frac{9}{16}\)
- 2: \(\frac{6}{16} = \frac{9}{x}\)
- 3: \(\frac{9}{6} = \frac{x}{16}\)
So, the correct answers are 1, 2, and 3.