Question

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1. 12 10 12 6 ?

Explanation:

Step1: Identify the shape and similarity

The figures are parallelograms, and they are similar (since the problem implies a proportional relationship). In similar figures, corresponding sides are proportional. The first parallelogram has sides 12 and 10, the second has one side 6 corresponding to 12? Wait, no, let's check the correspondence. Wait, the first parallelogram: two sides 12, two sides 10. The second parallelogram: one side 6, and we need to find the other side. Let's assume the sides are proportional. So the ratio of corresponding sides: 12 corresponds to 6? Wait, 12/6 = 2. Then the other side of the first is 10, so the corresponding side of the second would be 10/2 = 5? Wait, no, maybe the 10 corresponds to 6? Wait, no, let's see: the first parallelogram has sides 12 (length) and 10 (width). The second has side 6 (maybe corresponding to 12? Wait, 12 and 6: 12/6 = 2. So the other side (10) should be divided by 2? Wait, 10/2 = 5. Wait, maybe the proportion is 12/6 = 10/x? Wait, no, let's set up the proportion correctly. Let the unknown side be \( x \). If the first parallelogram has sides 12 and 10, and the second has sides 6 and \( x \), and they are similar, then the ratio of corresponding sides is equal. So \( \frac{12}{6} = \frac{10}{x} \)? Wait, no, maybe \( \frac{12}{10} = \frac{6}{x} \)? Wait, no, let's think again. The first parallelogram: two sides of length 12, two of length 10. The second: one side is 6, let's say it's corresponding to the side of length 12? Wait, 12 to 6 is a scale factor of \( \frac{6}{12} = \frac{1}{2} \). So the other side (10) would be scaled by \( \frac{1}{2} \)? No, wait, if the first has side 12 and the second has side 6, that's a reduction by half. So the other side of the first is 10, so the second should be 10 (6/12) = 5? Wait, no, 10 (6/12) = 5? Wait, 6/12 is 1/2, so 10 1/2 = 5. Alternatively, set up the proportion: \( \frac{12}{6} = \frac{10}{x} \)? Wait, no, cross-multiplying: 12x = 610 → 12x = 60 → x = 5. Yes, that works. So the unknown side is 5.

Step2: Solve the proportion

We set up the proportion between the corresponding sides of the similar parallelograms. Let the unknown side be \( x \). The ratio of the sides of length 12 and 6 should be equal to the ratio of the sides of length 10 and \( x \). So:

$$ \frac{12}{6} = \frac{10}{x} $$

Cross - multiply:

$$ 12x = 6\times10 $$

$$ 12x = 60 $$

Divide both sides by 12:

$$ x=\frac{60}{12}=5 $$

Answer:

5