Question

if these two shapes are similar, what is the measure of the missir 1 mm 3 mm 12 mm z = \square millimeters submit

Explanation:

Step1: Identify the ratio of corresponding sides

For similar triangles, the ratios of corresponding sides are equal. The small triangle has sides 1 mm and 3 mm, and the large triangle has sides \( z \) and 12 mm. The ratio of the sides of the small triangle to the large triangle is \( \frac{3}{12}=\frac{1}{4} \)? Wait, no, wait. Wait, the corresponding sides: the side of length 3 mm in the small triangle corresponds to 12 mm in the large triangle. So the scale factor is \( \frac{12}{3} = 4 \). Alternatively, the side of length 1 mm in the small triangle corresponds to \( z \) in the large triangle. So we set up the proportion \( \frac{1}{z}=\frac{3}{12} \).

Step2: Solve the proportion

Cross - multiply: \( 3z = 1\times12 \). Then \( z=\frac{12}{3}=4 \). Wait, no, wait. Wait, let's check again. The small triangle: side 1 mm and 3 mm. The large triangle: side \( z \) and 12 mm. Since they are similar, the ratio of corresponding sides is equal. So \( \frac{1}{z}=\frac{3}{12} \). Cross - multiplying gives \( 3z = 12\times1 \), so \( z=\frac{12}{3}=4 \). Wait, or maybe the other way: the ratio of the small to large is \( \frac{3}{12}=\frac{1}{4} \), so the side of length 1 mm in small corresponds to \( 1\times4 = 4 \) mm in large? Wait, no, wait. Wait, the side of length 3 mm in small is corresponding to 12 mm in large. So the scale factor from small to large is \( \frac{12}{3}=4 \). So the side of length 1 mm in small will correspond to \( 1\times4 = 4 \) mm in large. So \( z = 4 \).

Answer:

4