Understanding Similar Triangles and Perimeters

The question deals with two similar triangles, ΔABC and ΔDEF. We are given their perimeters and the length of one side in ΔDEF (DE = 6 cm). We need to find the length of the corresponding side in ΔABC (AB).

Similar triangles have corresponding angles that are equal and corresponding sides that are in proportion. A key property related to similar triangles is that the ratio of their perimeters is equal to the ratio of their corresponding sides.

Mathematically, if ΔABC ~ ΔDEF, then:

• $\angle\text{A} = \angle\text{D}$, $\angle\text{B} = \angle\text{E}$, $\angle\text{C} = \angle\text{F}$
• $\frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{AC}}{\text{DF}}$
• $\frac{\text{Perimeter}(\Delta\text{ABC})}{\text{Perimeter}(\Delta\text{DEF})} = \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{AC}}{\text{DF}}$

Applying the Perimeter Ratio Property

We are given:

• Perimeter of ΔABC = 40 cm
• Perimeter of ΔDEF = 12 cm
• DE = 6 cm

We want to find AB. Since AB is the side corresponding to DE (as the triangles are named ΔABC ~ ΔDEF, so A corresponds to D, B to E, C to F), we can use the ratio of perimeters and the ratio of these corresponding sides:

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

Calculating the Side Length AB

Now, let's substitute the given values into the equation:

$\frac{40\text{ cm}}{12\text{ cm}} = \frac{\text{AB}}{6\text{ cm}}$

To find AB, we can rearrange the equation:

$\text{AB} = \left(\frac{40}{12}\right) \times 6\text{ cm}$

Simplify the fraction $\frac{40}{12}$: both 40 and 12 are divisible by 4.

$\frac{40}{12} = \frac{40 \div 4}{12 \div 4} = \frac{10}{3}$

So the equation becomes:

$\text{AB} = \left(\frac{10}{3}\right) \times 6\text{ cm}$

Now, multiply:

$\text{AB} = \frac{10 \times 6}{3}\text{ cm}$

$\text{AB} = \frac{60}{3}\text{ cm}$

$\text{AB} = 20\text{ cm}$

Thus, the length of side AB is 20 cm.

Verification

Let's check if the ratio of sides matches the ratio of perimeters:

Ratio of perimeters = $\frac{40}{12} = \frac{10}{3}$

Ratio of corresponding sides $\frac{\text{AB}}{\text{DE}} = \frac{20}{6} = \frac{10}{3}$

The ratios match, confirming our calculation for AB is correct based on the properties of similar triangles.

Similar Triangle Properties Summary

• Corresponding angles are equal.
• Corresponding sides are proportional.
• Ratio of areas is the square of the ratio of corresponding sides.
• Ratio of perimeters is equal to the ratio of corresponding sides.

Revision Table: Similar Triangles Perimeter

Key concepts for solving similar triangle problems involving perimeters:

Additional Information: Scale Factor in Similar Triangles

The constant ratio between corresponding sides of similar triangles is called the scale factor. In this problem, the ratio of perimeters is $\frac{40}{12} = \frac{10}{3}$. This means the scale factor from ΔDEF to ΔABC is $\frac{10}{3}$, and from ΔABC to ΔDEF is $\frac{3}{10}$.

If you know the scale factor, you can find the length of any corresponding side. For example, if the scale factor from ΔDEF to ΔABC is $k$, then any side length in ΔABC is $k$ times the corresponding side length in ΔDEF.

Here, $k = \frac{10}{3}$. So, $\text{AB} = k \times \text{DE} = \frac{10}{3} \times 6 = 20$ cm. This matches our result.