Question

Prove: a line cd is a perpendicular bisector of another line ab. a triangle is drawn by the points of a, b, and c. complete the proof. statements reasons 1. is the perpendicular bisector of 1. given 2. d is the midpoint of 2. 3. 3. definition of a midpoint 4. 4. definition of congruence 5. 5. 6. and are right angles 6. definition of a perpendicular bisector 7. and right triangles 7. definition of a right triangle 8. 8. pythagorean theorem 9. 9. substitution property of equality 10. 10. substitution property of equality 11. 11.

Answer 1

To prove that line cd is a perpendicular bisector of line ab, follow these steps: show that line cd is the midpoint of line ab, demonstrate that line cd is perpendicular to line ab, and use the Pythagorean theorem to show that the sum of the squares of the lengths of the segments formed by cd and ab are equal.

Step-by-step explanation:

To prove that line cd is a perpendicular bisector of line ab, we can follow the given steps:

1. Given: Line cd is a perpendicular bisector of line ab.
2. D is the midpoint of AB (given).
3. By definition of a midpoint, AD = DB.
4. By definition of congruence, AD = CD and DB = CD.
5. Since AD = DB, CD is perpendicular to AB (by definition of a perpendicular bisector).
6. By definition of a right triangle, ΔCDA and ΔCDB are right triangles.
7. By the Pythagorean theorem, CA^2 + AD^2 = CD^2 and CB^2 + DB^2 = CD^2.
8. Use the substitution property of equality to equate CA^2 + AD^2 and CB^2 + DB^2.
9. Use the substitution property of equality again to equate CA^2 + AD^2 and CB^2 + DB^2 with CD^2.
10. Therefore, CA^2 + AD^2 = CB^2 + DB^2 = CD^2.