SOLUTION: Prove that if two intersecting line segments meet at their midpoints, the line segments joining their endpoints are congruent. Given: O is the midpoint of XY and of AB Prove: XA

Algebra.Com
Question 1192419: Prove that if two intersecting line segments meet at their midpoints, the line
segments joining their endpoints are congruent.
Given: O is the midpoint of XY and of AB
Prove: XA ≅ BY
Found 2 solutions by CPhill, Edwin McCravy:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
**1. Definition of Midpoint:**
* Since O is the midpoint of XY, we know that OX ≅ OY.
* Since O is the midpoint of AB, we know that OA ≅ OB.
**2. Vector Addition:**
* Express XA as the sum of vectors: XA = XO + OA
* Express BY as the sum of vectors: BY = BO + OY
**3. Substitute with Equal Vectors**
* Since OX = -OY and OA = -OB, we can substitute:
* XA = -OY + OA
* BY = -OA + OY
**4. Rearrange**
* XA = -OY + OA
* BY = OY - OA
* Observe that XA and BY are the same vector but with opposite signs.
**5. Conclusion**
* Since XA and BY have the same magnitude but opposite directions, they are congruent.
**Therefore, if two intersecting line segments meet at their midpoints, the line segments joining their endpoints are congruent.**
This proof utilizes vector concepts to demonstrate the congruence of the line segments.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
 
Given: O is the midpoint of XY and of AB
Prove: XA ≅ BY

 

1. XO ≅ YO       |1. O is the midpoint of XY
2. ∠XOA ≅ ∠YOB   |2. vertical angles are congruent   
3. AO ≅ BO       |3. O is the midpoint of AB
4. ΔXOA ≅ ΔYOB   |4. side-angle-side    
5. XA ≅ BY       |5. corresponding parts of congruent triangles

Edwin
RELATED QUESTIONS

a tri colored flag is made out of a rectangle piece of cloth whose corners are labeled a... (answered by robertb)
Given a quadrilateral ( -7, 2 ) ( -5, -8 ), ( 3, -9 ) and ( 4, -3 ) show that the line... (answered by ikleyn)
Prove that the line segments joining the midpoints of the sides of an equilateral... (answered by ewatrrr)
I have two problems that I need help: Problem #1: Prove triangle ABC is congruent... (answered by venugopalramana)
Hi, can you help me solve this problem? Given a quadrilateral whose vertices are ( 3, 5 (answered by ikleyn,Edwin McCravy)
How can I show that two line segments whose endpoints are (8,0) (6,6) and (5,7), and... (answered by ReadingBoosters)
Complete the formal proof of the theorem. If two line segments are congruent, then their (answered by CPhill,ikleyn)
prove the following using coordinate geometry: The segments joining the midpoints of the... (answered by Edwin McCravy)
1. State 3 points D, E and F, ensuring they are not all on the same line. 2. Find the... (answered by Solver92311)