Math Problem Statement

AE = EC and BF = FC. BA = 18, DE = 7, and BC = 14. What is the length of FE?

Solution

Let's solve the problem in the image.

Given Information:

  • ABC\triangle ABC is a triangle with D,E,FD, E, F as midpoints of sides AB,AC,AB, AC, and BCBC respectively.
  • AE=ECAE = EC (implies EE is the midpoint of ACAC).
  • BF=FCBF = FC (implies FF is the midpoint of BCBC).
  • BA=18BA = 18, DE=7DE = 7, and BC=14BC = 14.

Required:

Find the length of FEFE.


Key Concept: Midsegment Theorem

The midsegment theorem states that:

  1. A midsegment of a triangle (a segment connecting the midpoints of two sides) is parallel to the third side.
  2. The length of the midsegment is half the length of the side it is parallel to.

Application:

  1. Since FEFE is the midsegment connecting FF (midpoint of BCBC) and EE (midpoint of ACAC), it is parallel to side ABAB.
  2. By the midsegment theorem, the length of FEFE is: FE=12×ABFE = \frac{1}{2} \times AB

Substitution:

From the given information, AB=18AB = 18. Thus: FE=12×18=9FE = \frac{1}{2} \times 18 = 9


Final Answer:

FE=9FE = 9

Let me know if you'd like further clarification!


Related Questions:

  1. What are the other midsegments of ABC\triangle ABC, and how can their lengths be calculated?
  2. How does the midsegment theorem help in proving similarity between triangles?
  3. What happens to the triangle if all three midsegments are drawn? What new figure is formed?
  4. Can the midsegment theorem be applied to a right triangle? If yes, how?
  5. How can the perimeter of the smaller triangle formed by midsegments be calculated?

Tip:

When working with midsegments, always identify the side they are parallel to—it simplifies the problem significantly!

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Math Problem Analysis

Mathematical Concepts

Geometry
Midsegments of a Triangle
Proportionality

Formulas

Midsegment = 1/2 * Length of Parallel Side

Theorems

Midsegment Theorem

Suitable Grade Level

Grades 8-10