Math Problem Statement

Solution

Let's solve the given problem step by step. The problem provides the following information:

  1. DEABDE \parallel AB
  2. CFABCF \parallel AB
  3. BGABBG \parallel AB

The goal is to find the length of xx.

The diagrams show a set of similar triangles. We can use the properties of similar triangles to solve for xx.

Since DEABDE \parallel AB, CDE\triangle CDE is similar to CBA\triangle CBA.

Since CFABCF \parallel AB, CFB\triangle CFB is similar to CBA\triangle CBA.

And since BGABBG \parallel AB, BGA\triangle BGA is similar to CBA\triangle CBA.

Given: DC=7DC = 7 CB=3CB = 3 BA=xBA = x

First, let's write the similarity ratios: For CDECBA\triangle CDE \sim \triangle CBA: DCCB=DEBA\frac{DC}{CB} = \frac{DE}{BA}

For CFBCBA\triangle CFB \sim \triangle CBA: CFBA=CBBA\frac{CF}{BA} = \frac{CB}{BA}

For BGACBA\triangle BGA \sim \triangle CBA: BGBA=BABA\frac{BG}{BA} = \frac{BA}{BA}

To find xx: 73=7x\frac{7}{3} = \frac{7}{x}

Since DEABDE \parallel AB, all segments are proportional. So, the ratio will be:

73=DEx\frac{7}{3} = \frac{DE}{x}

The problem becomes:

7x=377 \cdot x = 3 \cdot 7 7x=217x = 21 x=3x = 3

However, this doesn't align with the problem constraints. To address this, we must correct the interpretation, but let's consider if the proper interpretation involves additional triangles:

Correct interpretation might involve:

Considering intermediate steps and correct proportions: DECF=CFBG\frac{DE}{CF} = \frac{CF}{BG}

This forms a valid proportional statement: 73+x=3x\frac{7}{3 + x} = \frac{3}{x} By cross-multiplying: 7x=3(3+x)7x = 3(3 + x) 7x=9+3x7x = 9 + 3x 7x3x=97x - 3x = 9 4x=94x = 9 x=942.25x = \frac{9}{4} \approx 2.25

Hence, based on updated triangles and correctly solving proportion: x=94x = \frac{9}{4}

Tip: Always confirm proportional relationships to avoid basic misinterpretations.

Further Questions:

  1. What are properties of similar triangles?
  2. Explain AA (Angle-Angle) similarity criterion?
  3. Discuss properties of parallel lines?
  4. What is cross-multiplication used for?
  5. Explain ratios and proportions in triangles?
  6. Define parallel line segment properties?
  7. What is the method for solving proportion equations?
  8. Discuss key steps for validating triangle similarity?

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Math Problem Analysis

Mathematical Concepts

Geometry
Similar Triangles
Proportions

Formulas

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Theorems

AA (Angle-Angle) similarity criterion

Suitable Grade Level

Grades 9-12