Notes on Lines, line segments, and rays. Notes on Properties if planes, lines, and points. Notes on Describe intersections in a plane. Notes on Additive property of lengths. Notes on congruent line segments. Notes on Construct a congruent segment. Notes on Distance formula. Notes on Construct the midpoint or perpendicular bisector of a segment. Notes on Midpoint. Notes on angle vocabulary. Notes on Construct an angle bisector. Notes on Construct a congruent angle. Notes on parts of a circle. Notes on arc length. Notes on slopes of parallel and perpendicular lines. Notes on equations of parallel and perpendicular lines. Notes on find the distance between a point and a line. Notes on find the distance between two parallel lines. Notes on Construct an equilaterial triangle inscribed in a circle. Notes on Construct a square inscribed in a circle. Notes on Construct a regular hexagon inscribed in a circle. Notes on Construct a equilaterial triangle or regular hexagon. Notes on Construct a square. Notes on translations write the rule. Notes on Reflections graph the image. Notes on Reflections find the coordinates. Notes on rotate polygons about a point. Notes on rotations graph the image. Notes on rotations find the coornates. Make Notes detailed on paper all notes on each topic detailed throughly.Notes on Lines, line segments, and rays. Notes on Properties if planes, lines, and points. Notes on Describe intersections in a plane. Notes on Additive property of lengths. Notes on congruent line segments. Notes on Construct a congruent segment. Notes on Distance formula. Notes on Construct the midpoint or perpendicular bisector of a segment. Notes on Midpoint. Notes on angle vocabulary. Notes on Construct an angle bisector. Notes on Construct a congruent angle. Notes on parts of a circle. Notes on arc length. Notes on slopes of parallel and perpendicular lines. Notes on equations of parallel and perpendicular lines. Notes on find the distance between a point and a line. Notes on find the distance between two parallel lines. Notes on Construct an equilaterial triangle inscribed in a circle. Notes on Construct a square inscribed in a circle. Notes on Construct a regular hexagon inscribed in a circle. Notes on Construct a equilaterial triangle or regular hexagon. Notes on Construct a square. Notes on translations write the rule. Notes on Reflections graph the image. Notes on Reflections find the coordinates. Notes on rotate polygons about a point. Notes on rotations graph the image. Notes on rotations find the coornates. Make Notes detailed on paper all notes on each topic detailed throughly.Notes on Lines, line segments, and rays. Notes on Properties if planes, lines, and points. Notes on Describe intersections in a plane. Notes on Additive property of lengths. Notes on congruent line segments. Notes on Construct a congruent segment. Notes on Distance formula. Notes on Construct the midpoint or perpendicular bisector of a segment. Notes on Midpoint. Notes on angle vocabulary. Notes on Construct an angle bisector. Notes on Construct a congruent angle. Notes on parts of a circle. Notes on arc length. Notes on slopes of parallel and perpendicular lines. Notes on equations of parallel and perpendicular lines. Notes on find the distance between a point and a line. Notes on find the distance between two parallel lines. Notes on Construct an equilaterial triangle inscribed in a circle. Notes on Construct a square inscribed in a circle. Notes on Construct a regular hexagon inscribed in a circle. Notes on Construct a equilaterial triangle or regular hexagon. Notes on Construct a square. Notes on translations write the rule. Notes on Reflections graph the image. Notes on Reflections find the coordinates. Notes on rotate polygons about a point. Notes on rotations graph the image. Notes on rotations find the coornates. Make Notes detailed on paper all notes on each topic detailed throughly

Lines, Line Segments, and Rays

• Line: A straight path that extends infinitely in both directions. Represented with arrows on both ends (e.g., line AB).

• Line Segment: A part of a line that has two endpoints. It is finite in length (e.g., segment AB).

• Ray: A part of a line that starts at one point and extends infinitely in one direction (e.g., ray AB starts at A and goes through B).

Properties of Planes, Lines, and Points

• Point: A location in space with no dimensions or size.

• Line: Infinite set of points extending in two directions.

• Plane: A flat surface that extends infinitely in two dimensions and is defined by three non-collinear points.

• Properties:

  • A line is defined by any two points.

  • A plane is determined by three non-collinear points.

Describe Intersections in a Plane

• Intersection: The point or points where two lines, segments, or planes meet in a plane.

• Types of Intersections:

  • Coincident: Lines that lie on top of each other.

  • Parallel: Lines that never intersect.

  • Intersecting: Lines that cross at one point.

Additive Property of Lengths

• If a point B lies on line segment AC, then the length of AB + length of BC = length of AC.

Congruent Line Segments

• Definition: Line segments that have the same length.

• Notation: If AB ≅ CD, then the lengths of AB and CD are equal.

Constructing a Congruent Segment

1. Draw the segment you want to replicate.

2. Use a compass to measure the length of this segment.

3. Place the compass point at the new starting point and draw an arc.

4. Mark the intersection of the arc with a straight line as the endpoint.

Distance Formula

• The distance d between two points (x1, y1) and (x2, y2) is given by:
  $d = ext{sqrt}((x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2)$

Constructing the Midpoint or Perpendicular Bisector of a Segment

• Midpoint: The point that divides the segment into two equal parts.

  1. Find the coordinates of the endpoints.

  2. Use the midpoint formula:
     $M = \left(\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2}\right)$

• Perpendicular Bisector: A line that is perpendicular to a segment at its midpoint.

Midpoint

• Defined as the average of the coordinates of the endpoints. For segment AB:
  $M = \left(\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2}\right)$

Angle Vocabulary

• Angle: Formed by two rays with a common endpoint, called the vertex.

• Types of Angles:

  • Acute: Less than 90°.

  • Right: Exactly 90°.

  • Obtuse: More than 90° but less than 180°.

  • Straight: Exactly 180°.

Constructing an Angle Bisector

1. Draw the angle.

2. Use a compass to draw arcs from both rays.

3. Mark the intersection points.

4. Draw a line from the vertex through the intersection to bisect the angle.

Constructing a Congruent Angle

1. Draw the original angle.

2. Using a compass, replicate the angle size by measuring the radius of the arc.

3. Mark the new vertex.

4. Draw rays to form the congruent angle.

Parts of a Circle

• Center: The point within the circle.

• Radius: Distance from the center to a point on the circle.

• Diameter: A line segment that passes through the center and has endpoints on the circle (diameter = 2 × radius).

• Circumference: The total distance around the circle, calculated as $C = 2 ext{π}r$.

Arc Length

• The distance between two points on a circle along the circumference; calculated as:
  $ext{Arc Length} = rac{ heta}{360} imes 2 ext{π}r$ (where $heta$ is the angle in degrees).

Slopes of Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope (m1 = m2).

• Perpendicular Lines: Slopes are negative reciprocals (m1 imes m2 = -1).

Equations of Parallel and Perpendicular Lines

• General Form: $y = mx + b$.

• For parallel lines: Keep the same slope.

• For perpendicular lines: If line 1 has slope m1, then line 2 has slope $m<em>2 = -\frac{1}{m</em>1}$.

Finding the Distance Between a Point and a Line

• Use the formula:
  $d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}$, where Ax + By + C = 0 is the line equation.

Finding the Distance Between Two Parallel Lines

• The distance between lines of the form:
  $Ax + By + C<em>1 = 0$ $Ax + By + C</em>2 = 0$ is given by:
  $d = \frac{|C<em>2 - C</em>1|}{\sqrt{A^2 + B^2}}$.

Constructing an Equilateral Triangle Inscribed in a Circle

1. Draw a circle.

2. Mark a point on the circle for the first vertex.

3. Use a compass to measure the radius and mark two additional points on the circumference.

4. Connect the points to form an equilateral triangle.

Constructing a Square Inscribed in a Circle

1. Draw a circle.

2. Draw two perpendicular diameters to identify the midpoints.

3. Connect these points to form a square.

Constructing a Regular Hexagon Inscribed in a Circle

1. Draw a circle.

2. Mark 6 points at equal distances around the circle (angle of 60° apart).

3. Connect these points to form a hexagon.

Constructing an Equilateral Triangle or Regular Hexagon

• Follow the same steps described in previous sections for constructing triangles and hexagons.

Constructing a Square

• Similar to the method for constructing a square inscribed in a circle but without specific points on a circle.

Translations: Write the Rule

• A transformation that slides a figure in a straight line from one position to another without changing its shape, size, or orientation. Rule: $T(x, y) = (x + a, y + b)$.

Reflections: Graph the Image

• A transformation across a line (the line of reflection) that creates a mirror image.

Reflections: Find the Coordinates

• For a point (x, y) reflected over a line (y = mx + b), use the formula for finding the new coordinates after reflection based on the slope.

Rotate Polygons About a Point

• A transformation that turns a polygon around a fixed point (the center of rotation). The angle of rotation and direction (clockwise or counterclockwise) determine the new position.

Rotations: Graph the Image

• Graph the polygon after rotation using the angle and direction of rotation.

Rotations: Find the Coordinates

• Use the rotation formulas for each vertex based on the given angle and center of rotation.