Concepts
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Points and Lines
- Point: An exact location in space, usually labelled with a capital letter (e.g., A).
- Line: Extends forever in both directions; made up of infinite points.
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Line Segments and Rays
- Line Segment: Part of a line with two endpoints.
- Ray: Part of a line that starts at one point and extends forever in one direction.
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Measuring Angles
- Measured in degrees (°).
- Angles are formed when two rays share a common endpoint (vertex).
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Types of Angles
- Acute: Less than 90°
- Right: Exactly 90°
- Obtuse: Greater than 90° but less than 180°
- Straight: Exactly 180°
- Reflex: Greater than 180° but less than 360°
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Triangles
- 3 sides, 3 angles. Types: equilateral, isosceles, scalene, right, acute, obtuse.
- Sum of angles: 180°
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Quadrilaterals
- 4 sides. Types: square, rectangle, parallelogram, rhombus, trapezoid, kite.
- Sum of angles: 360°
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Polygons
- Closed shapes with 3 or more straight sides.
- Regular: all sides and angles equal. Irregular: not all sides/angles equal.
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Circles
- Set of all points equidistant from the center.
- Parts: center, radius, diameter, chord, arc, sector, circumference.
- Circumference: \(C = 2\pi r\)
- Area: \(A = \pi r^2\)
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Solid Figures
- 3D objects: cube, rectangular prism, sphere, cylinder, cone, pyramid.
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Surface Area
- The total area of all faces of a solid figure.
- Cube: \(6a^2\); Rectangular prism: \(2(lw + lh + wh)\)
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Graphing Figures
- Placing points, lines, and shapes on a coordinate plane (x, y axes).
- Each point is an ordered pair (x, y).
Examples
Example 1: Naming a Ray
Question: Name the ray that starts at point A and passes through B.
Solution: Ray AB (not BA).
Question: Name the ray that starts at point A and passes through B.
Solution: Ray AB (not BA).
Example 2: Measuring an Angle
Question: What is the measure of a right angle?
Solution: 90°
Question: What is the measure of a right angle?
Solution: 90°
Example 3: Types of Triangles
Question: A triangle has angles 90°, 45°, and 45°. What type is it?
Solution: Right isosceles triangle
Question: A triangle has angles 90°, 45°, and 45°. What type is it?
Solution: Right isosceles triangle
Example 4: Area of a Circle
Question: What is the area of a circle with \(r=3\)?
Solution: \(A = \pi r^2 = \pi \times 9 \approx 28.27\)
Question: What is the area of a circle with \(r=3\)?
Solution: \(A = \pi r^2 = \pi \times 9 \approx 28.27\)
Example 5: Surface Area of a Cube
Question: If a cube has edge 2 units, what is its surface area?
Solution: \(6 \times 2^2 = 24\) square units
Question: If a cube has edge 2 units, what is its surface area?
Solution: \(6 \times 2^2 = 24\) square units