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 , the bisectors of angles and meet the opposite sides at points and , respectively. We are tasked with proving that .
Step-by-Step Solution:
-
Given:
- is isosceles, with .
- The angle bisectors of and meet the opposite sides and at points and , respectively.
- We need to prove that .
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Draw the figure: Start by drawing triangle such that , and then draw the bisectors of and , meeting sides and at points and , respectively.
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Use the Angle Bisector Theorem: According to the angle bisector theorem:
- Since , it follows that . Therefore, point is the midpoint of .
Similarly, applying the angle bisector theorem at :
- Since , . Thus, point is the midpoint of .
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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 is the midpoint of and is the midpoint of . Therefore, by the midpoint theorem, and .
Conclusion:
We have proven that by using the midpoint theorem, based on the properties of angle bisectors and the fact that .
Would you like further clarification on any part of the solution?
Here are 5 related questions:
- What is the angle bisector theorem, and how is it applied in various triangles?
- How do you prove the midpoint theorem in a triangle?
- Can the midline property be applied to non-isosceles triangles? Why or why not?
- How do congruent triangles help in proving parallel lines in geometrical problems?
- 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
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