Question

9 x 10 54 30 60 scale factor= __ missing side length = __

Explanation:

Step1: Find Scale Factor

Compare corresponding sides (e.g., 54 and 9). Scale factor = $\frac{54}{9} = 6$ (or $\frac{60}{10}=6$, $\frac{30}{x}$ should also be 6).

Step2: Solve for Missing Side \( x \)

Using scale factor 6: $\frac{30}{x} = 6$ → \( x = \frac{30}{6} = 5 \)? Wait, no—wait, first triangle is smaller, second is larger. Wait, 9 to 54: 9×6=54, 10×6=60, so x×6=30 → \( x = \frac{30}{6} = 5 \)? Wait, original first triangle: sides 9, 10, x; second: 54, 60, 30. So scale factor from first to second is 6 (since 9×6=54, 10×6=60), so x×6=30 → x=5. Wait, maybe I mixed up. Wait, scale factor is second/first: 54/9=6, 60/10=6, so 30/x=6 → x=30/6=5. Wait, but let's check again.

Wait, maybe the first triangle is the smaller one, second is larger. So scale factor (larger/smaller) is 54/9=6, 60/10=6, so 30/x=6 → x=5. Wait, but the initial wrong answer was 5? Wait, the user's handwritten was 5 and 1.80, which is wrong. Let's correct.

Step1: Determine Scale Factor

Take corresponding sides: 54 (second) and 9 (first). Scale factor \( k = \frac{54}{9} = 6 \). (Or 60/10=6, consistent.)

Step2: Find Missing Side \( x \)

The side \( x \) in the first triangle corresponds to 30 in the second. So \( x \times k = 30 \) → \( x = \frac{30}{k} = \frac{30}{6} = 5 \). Wait, no—wait, if first is smaller, then second is larger, so first side × scale factor = second side. So 9×6=54, 10×6=60, so x×6=30 → x=5. Yes, that's correct. The handwritten 1.80 was wrong; it should be 5. And scale factor is 6 (not 5, the handwritten 5 was wrong).

Wait, let's re-express:

Scale Factor (from small to large) = 54/9 = 6.

Missing side \( x \): since 30 is the large side, small side \( x = 30 / 6 = 5 \).

Answer:

Scale Factor = \(\boldsymbol{6}\)
Missing Side length = \(\boldsymbol{5}\)