lecture recording on 30 October 2024 at 19.29.59 PM

Chapter 1: Introduction to Midpoints

• Definition of Midpoint: A midpoint is a point that divides a segment into two congruent segments.

• Examples:

  • Point A: Not a midpoint (located on a line, not a segment).

  • Point B: Midpoint (divides the segment into two congruent segments).

  • Point C: Not a midpoint (located on a ray, not a segment).

• Illustration: For segment RS with ray AB passing through at point X (where RX=XS=5), point X is identified as the midpoint of segment RS because half segments are congruent.

Chapter 2: Bisector of Angle

• Definition of Segment Bisector: A ray that divides a segment into two congruent parts.

• Example: Ray AB bisects RS into two congruent parts.

• Angle Bisector:

  • An angle bisector is a ray that divides an angle into two congruent angles.

  • For instance, if angle ABC = 60 degrees, then BD as an angle bisector creates two angles (ABD and DBC), each measuring 30 degrees.

Chapter 3: Midpoint of Segment Problems

• Problem Example: If B is the midpoint of segment AC, with BC = 5x - 8 and AB = 2x + 4,

  • Since B is the midpoint, AB = BC (congruent segments).

  • Thus, 2x + 4 = 5x - 8.

• Solving:

  1. Rearrange: Subtract 2x from both sides (3x = 12) and add 8 to both sides (4 + 8 = 12).

  2. Divide by 3 (x = 4).

  3. Compute AB = 2x+4 = 12 and BC = 12. Add to find length of AC = 24.

Chapter 4: Measure of Angle Problems

• Theoretical Procedures: When a ray BD bisects angle ABC, find angle ABD if ABC = 11x + 5 and DBC = 5x + 6.

• Steps:

  1. Sketch angle ABC and ray BD.

  2. By definition of angle bisector, ABD = DBC.

  3. Set equations: 11x + 5 = 2(5x + 6).

• Calculation:

  • Distributing yields: 10x + 12 = 11x + 5.

  • Solve for x: 12 - 5 = x (x = 7).

Chapter 5: Solve for ABD and DBC

• Calculate:Find angle DBC using x value: 5x + 6.

  • When x = 7, DBC = 35 + 6 = 41.

  • Therefore, angle ABD = 41 degrees, which is the answer to the problem.

Chapter 6: Midpoint Segment Ratios

• Example: If B is midpoint of segment AD, and C lies between B and D, determine length of segment AD given the ratio BC to CD as 3:7 with BC = 15 units.

• Solution Process:

  1. If BC relates as 3x and CD as 7x, then BC + CD = BD, hence BD = 10x.

  2. By knowing BC = 15 (3x = 15), solve x = 5.

• Finding Lengths:

  • BC = 15, CD = 7x = 35, and BD = 10x = 50 and hence AD = CB + BD = 100 units long.

Chapter 7: Conclusion of Findings

• The calculations demonstrated rigor in applying the concepts of midpoints and angle bisectors to derive lengths and angular measures through algebraic methods.

I cannot create drawings, but I can guide you on what illustrations to add. For example:

• For midpoints, draw a line segment with clearly marked endpoints and indicate the midpoint.

• For angle bisectors, draw an angle with a ray dividing it into two equal parts.

• For the segment ratios, you can depict a segment divided into two parts showing the lengths in relation to the total segment length.