Question

Consider the enlargement of the triangle.

A smaller triangle with side lengths 9 millimeters and 6 millimeters. A larger triangle with side lengths x millimeters and 16 millimeters.

Which proportional statements are true for finding the value of x? Check all that apply.
StartFraction 6 over x EndFraction = StartFraction 9 over 16 EndFraction
StartFraction 6 over 16 EndFraction = StartFraction 9 over x EndFraction
StartFraction 9 over 6 EndFraction = StartFraction x over 16 EndFraction
StartFraction 9 over x EndFraction = StartFraction 16 over 6 EndFraction
StartFraction 9 over x EndFraction = StartFraction 6 over 16 EndFraction

Answer 1

In similar triangles, the ratios of corresponding side lengths are equal. You can set up proportional statements based on the given side lengths of the smaller and larger triangles.

The smaller triangle has side lengths of 9 mm and 6 mm, while the larger triangle has corresponding side lengths of \( x \) mm and 16 mm.

To find the correct proportional statements, we can use the following relationships:

1. Corresponding side ratios should be equal:
\(\frac{6}{x} = \frac{9}{16}\)
\(\frac{9}{16} = \frac{x}{6}\)
\(\frac{x}{16} = \frac{9}{6}\)
\(\frac{9}{x} = \frac{16}{6}\)
\(\frac{9}{x} = \frac{6}{16}\)

Now let's consider each of the provided statements:

1. **\(\frac{6}{x} = \frac{9}{16}\)**: True
2. **\(\frac{6}{16} = \frac{9}{x}\)**: True (this can be cross-multiplied to verify)
3. **\(\frac{9}{6} = \frac{x}{16}\)**: True (also can be cross-multiplied)
4. **\(\frac{9}{x} = \frac{16}{6}\)**: True (can be verified by cross-multiplying)
5. **\(\frac{9}{x} = \frac{6}{16}\)**: True (can be verified by cross-multiplying)

Thus, **all of the listed proportional statements are true** for finding the value of \( x \).