Question

7. Proving the Perpendicular Bisector Theorem Using Reflections

Given: Point P is on the perpendicular bisector m of AB.
Prove: PA PB

Answer 1

To prove the Perpendicular Bisector Theorem, which states that a point on the perpendicular bisector of a segment is equidistant from the segment's endpoints, we use reflections and congruent triangles to show that PA is equal to PB.

Step-by-step explanation:

To prove the Perpendicular Bisector Theorem using reflections, we start with the given that point P is on the perpendicular bisector m of segment AB. Our goal is to prove that the lengths of PA and PB are equal.

Steps to Prove PA = PB

1. 
   
3. Let m be the perpendicular bisector of segment AB. This means m intersects AB at its midpoint, say M, and is perpendicular to AB.
4. 
   
6. Point P is given to be on line m; therefore, by the definition of a perpendicular bisector, PM is perpendicular to AB and P is equidistant from A and B.
7. 
   
9. Reflect triangle PAM over line m to get triangle PBM. Since reflections preserve distance, PM is congruent to itself (reflexive property), and AM is congruent to BM (since M is the midpoint of AB).
10. 
    
12. By the Side-Angle-Side (SAS) congruence postulate for triangles, we can conclude that triangle PAM is congruent to triangle PBM.
13. 
    
15. Since corresponding parts of congruent triangles are congruent (CPCTC), it follows that PA is congruent to PB. This proves the lengths PA and PB are equal.

Therefore, we have proved that point P is equidistant from points A and B when it lies on the perpendicular bisector of AB, which is the essence of the Perpendicular Bisector Theorem.

Answer 2

6+3 = 9
Step-by-step explanation:

-3-6 is equivalent to 6-(-3). Subtracting a negative number is the same as adding a positive number, so 6-(-3) is the same as 6+3. 6+3 = 9.