Understanding Similar Triangles and Perimeter Ratios

The question involves two similar triangles, ΔDEF and ΔPQR. Similar triangles have the property that their corresponding angles are equal, and their corresponding sides are in proportion. This constant ratio between corresponding sides is also equal to the ratio of their perimeters, semi-perimeters, altitudes, medians, etc.

The symbol Δ indicates a triangle. So, ΔDEF is similar to ΔPQR means that triangle DEF has the same shape as triangle PQR, though they might be different in size.

We are given that the ratio of the semi-perimeter of ΔDEF to the semi-perimeter of ΔPQR is 4 : 5.

A semi-perimeter of a triangle is half of its perimeter. The perimeter is the sum of the lengths of its sides.

Relating Semi-perimeter Ratio to Side Ratio in Similar Triangles

A key property of similar triangles is that the ratio of their corresponding sides is equal to the ratio of their perimeters and also equal to the ratio of their semi-perimeters.

Given ΔDEF is similar to ΔPQR, their corresponding sides are DE and PQ, EF and QR, and FD and RP.

The ratio of corresponding sides is therefore:

\(\frac{\text{DE}}{\text{PQ}} = \frac{\text{EF}}{\text{QR}} = \frac{\text{FD}}{\text{RP}}\)

And this ratio is equal to the ratio of their semi-perimeters:

\(\frac{\text{DE}}{\text{PQ}} = \frac{\text{EF}}{\text{QR}} = \frac{\text{FD}}{\text{RP}} = \frac{\text{Semi-perimeter of } \Delta \text{DEF}}{\text{Semi-perimeter of } \Delta \text{PQR}}\)

We are given that \(\frac{\text{Semi-perimeter of } \Delta \text{DEF}}{\text{Semi-perimeter of } \Delta \text{PQR}} = \frac{4}{5}\).

So, we can write the relationship between corresponding sides and the semi-perimeter ratio as:

\(\frac{\text{DE}}{\text{PQ}} = \frac{4}{5}\)

Solving for the Length of DE

We are given the length of the side PQ as 15 cm. We need to find the length of the side DE.

Using the equation derived from the property of similar triangles:

\(\frac{\text{DE}}{15 \text{ cm}} = \frac{4}{5}\)

To find the value of DE, we can solve this proportion. Multiply both sides of the equation by 15:

\(\text{DE} = \frac{4}{5} \times 15 \text{ cm}\)

Now, perform the multiplication:

\(\text{DE} = 4 \times \frac{15}{5} \text{ cm}\)

\(\text{DE} = 4 \times 3 \text{ cm}\)

\(\text{DE} = 12 \text{ cm}\)

Conclusion: The Length of DE

The length of the side DE in ΔDEF is found to be 12 cm, based on the similarity of the triangles and the given ratio of their semi-perimeters and the length of PQ.

Revision Table: Properties of Similar Triangles

Additional Information on Triangle Similarity

Two triangles are similar if they satisfy any of the following criteria:

• AA Similarity: If two angles of one triangle are respectively equal to two angles of another triangle.
• SSS Similarity: If the lengths of the corresponding sides of two triangles are proportional.
• SAS Similarity: If an angle of one triangle is equal to an angle of the other triangle and the sides including these angles are proportional.

In this problem, the fact that the triangles are similar (ΔDEF is similar to ΔPQR) is given directly. We then use the property relating the ratio of semi-perimeters to the ratio of corresponding sides to solve for the unknown side length DE.

Remember, the ratio of perimeters is simply twice the ratio of semi-perimeters, so using either ratio results in the same scale factor for the sides.