Math Problem Statement

Use side side side method

Solution

To solve the problem using the Side-Side-Side (SSS) similarity method, we will determine the ratio of the corresponding sides of triangles ABC\triangle ABC and DEF\triangle DEF and then use this ratio to find the side lengths of DEF\triangle DEF.

Step 1: Calculate the side lengths of ABC\triangle ABC

  • AC=3cmAC = 3 \, \text{cm}
  • CB=4cmCB = 4 \, \text{cm}
  • AB=5cmAB = 5 \, \text{cm}

Step 2: Determine the scale factor

Given that ABC\triangle ABC and DEF\triangle DEF are similar, the ratio of the sides of DEF\triangle DEF to ABC\triangle ABC is consistent for all three sides.

Step 3: Calculate the area of ABC\triangle ABC

As previously calculated, the area of ABC\triangle ABC is: Area of ABC=6cm2\text{Area of } \triangle ABC = 6 \, \text{cm}^2

Step 4: Use the area to find the scale factor

Given the area of DEF\triangle DEF is 26.46cm226.46 \, \text{cm}^2, we find the scale factor kk as follows:

Area ratio=Area of DEFArea of ABC=26.4664.41\text{Area ratio} = \frac{\text{Area of } \triangle DEF}{\text{Area of } \triangle ABC} = \frac{26.46}{6} \approx 4.41 k2=4.41k=4.412.1k^2 = 4.41 \quad \Rightarrow \quad k = \sqrt{4.41} \approx 2.1

Step 5: Use the scale factor to find the side lengths of DEF\triangle DEF

Since the triangles are similar, the side lengths of DEF\triangle DEF are kk times the corresponding side lengths of ABC\triangle ABC.

DF=k×AC=2.1×3=6.3cmDF = k \times AC = 2.1 \times 3 = 6.3 \, \text{cm} FE=k×CB=2.1×4=8.4cmFE = k \times CB = 2.1 \times 4 = 8.4 \, \text{cm} DE=k×AB=2.1×5=10.5cmDE = k \times AB = 2.1 \times 5 = 10.5 \, \text{cm}

Answer:

The side lengths of DEF\triangle DEF using the SSS method are:

  • DF=6.3cmDF = 6.3 \, \text{cm}
  • FE=8.4cmFE = 8.4 \, \text{cm}
  • DE=10.5cmDE = 10.5 \, \text{cm}

Would you like more details on how to apply the SSS similarity method or have any other questions?

Here are some related questions you might find useful:

  1. How does the SSS similarity theorem differ from other similarity theorems?
  2. Can we determine the angles of DEF\triangle DEF given this information?
  3. How would the result change if the triangles were not similar?
  4. What is the difference between congruence and similarity in triangles?
  5. How can we apply the SSS method to other types of geometric figures?

Tip: The SSS similarity method is powerful because it only requires the side lengths of the triangles to determine if they are similar. Make sure the corresponding sides are correctly identified before applying the method.

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Math Problem Analysis

Mathematical Concepts

Geometry
Similarity of Triangles

Formulas

Ratio of areas in similar triangles
Scale factor in similar triangles

Theorems

Side-Side-Side (SSS) similarity theorem

Suitable Grade Level

Grades 9-12