Math Problem Statement

In the issoseles tri. ABC AB = BC The bisectors of angle B and angle C meet the opposite sides at D and E respectively. Prove that DE||BC.Solve using 9th grade maths

Solution

In the given isosceles triangle ABC, where AB=BCAB = BC, the bisectors of angles BB and CC meet the opposite sides at points DD and EE, respectively. We are tasked with proving that DEBCDE \parallel BC.

Step-by-Step Solution:

  1. Given:

    • ABC\triangle ABC is isosceles, with AB=BCAB = BC.
    • The angle bisectors of B\angle B and C\angle C meet the opposite sides ACAC and ABAB at points DD and EE, respectively.
    • We need to prove that DEBCDE \parallel BC.
  2. Draw the figure: Start by drawing triangle ABCABC such that AB=BCAB = BC, and then draw the bisectors of B\angle B and C\angle C, meeting sides ACAC and ABAB at points DD and EE, respectively.

  3. Use the Angle Bisector Theorem: According to the angle bisector theorem:

    • ADDC=ABBC\frac{AD}{DC} = \frac{AB}{BC}
    • Since AB=BCAB = BC, it follows that AD=DCAD = DC. Therefore, point DD is the midpoint of ACAC.

    Similarly, applying the angle bisector theorem at CC:

    • AEEB=ACBC\frac{AE}{EB} = \frac{AC}{BC}
    • Since AB=BCAB = BC, AE=EBAE = EB. Thus, point EE is the midpoint of ABAB.
  4. Properties of Midsegments in a Triangle:

    • In any triangle, the line segment joining the midpoints of two sides is parallel to the third side and is half of its length.
    • From the earlier steps, we know that DD is the midpoint of ACAC and EE is the midpoint of ABAB. Therefore, by the midpoint theorem, DEBCDE \parallel BC and DE=12BCDE = \frac{1}{2} BC.

Conclusion:

We have proven that DEBCDE \parallel BC by using the midpoint theorem, based on the properties of angle bisectors and the fact that AB=BCAB = BC.

Would you like further clarification on any part of the solution?

Here are 5 related questions:

  1. What is the angle bisector theorem, and how is it applied in various triangles?
  2. How do you prove the midpoint theorem in a triangle?
  3. Can the midline property be applied to non-isosceles triangles? Why or why not?
  4. How do congruent triangles help in proving parallel lines in geometrical problems?
  5. What are the differences between angle bisectors and perpendicular bisectors in a triangle?

Tip: Understanding the properties of special triangles, such as isosceles and equilateral triangles, simplifies solving many geometry problems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Angle Bisectors
Midline Theorem

Formulas

Angle Bisector Theorem: AD/DC = AB/BC
Midline Theorem: Line joining midpoints of two sides is parallel to the third side and half its length

Theorems

Angle Bisector Theorem
Midline Theorem

Suitable Grade Level

Grade 9