Relating Perimeters and Sides of Similar Triangles

The question asks us to find the length of a side (AB) in one triangle (ΔABC) given information about a similar triangle (ΔDEF). We know both triangles are similar (indicated by the '~' symbol), we have their perimeters, and the length of one side of ΔDEF.

The Property of Similar Triangles' Perimeters

A key concept in geometry is that for any two similar triangles, the ratio of their perimeters is exactly the same as the ratio of any pair of corresponding sides.

If we have two similar triangles, say ΔABC ~ ΔDEF, this property can be written as:

$$ \frac{\text{Perimeter of } \Delta ABC}{\text{Perimeter of } \Delta DEF} = \frac{\text{Side AB}}{\text{Corresponding Side DE}} = \frac{\text{Side BC}}{\text{Corresponding Side EF}} = \frac{\text{Side AC}}{\text{Corresponding Side DF}} $$

Using the Given Information

Let's list the values provided in the question:

• The triangles are similar: ΔABC ~ ΔDEF
• Perimeter of ΔABC = 32 cm
• Perimeter of ΔDEF = 12 cm
• Length of side DE = 6 cm

We need to find the length of side AB. AB is the side corresponding to DE because of the order the vertices are listed in the similarity statement (ΔABC ~ ΔDEF).

Setting Up the Equation

Using the property mentioned earlier, we can set up a ratio involving the perimeters and the corresponding sides AB and DE:

$$ \frac{\text{Perimeter}(\Delta ABC)}{\text{Perimeter}(\Delta DEF)} = \frac{AB}{DE} $$

Now, we substitute the given values into this equation:

$$ \frac{32 \text{ cm}}{12 \text{ cm}} = \frac{AB}{6 \text{ cm}} $$

Solving for the Length of AB

To find the length of AB, we need to isolate AB in the equation. We can do this by multiplying both sides of the equation by 6 cm:

$$ AB = \frac{32 \text{ cm}}{12 \text{ cm}} \times 6 \text{ cm} $$

Now, we can perform the calculation. Notice that 6 cm is half of 12 cm:

$$ AB = 32 \times \left( \frac{6}{12} \right) \text{ cm} $$

Simplify the fraction $$ \frac{6}{12} $$:

$$ \frac{6}{12} = \frac{1}{2} $$

Substitute this back into the equation for AB:

$$ AB = 32 \times \frac{1}{2} \text{ cm} $$

$$ AB = 16 \text{ cm} $$

Conclusion

By applying the property that the ratio of perimeters of similar triangles equals the ratio of their corresponding sides, we calculated that the length of side AB is 16 cm.