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

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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