Overview
This lecture reviews key geometry concepts and formulas needed for SAT, ACT, and final exams, focusing on circles, squares, rectangles, and right triangles.
Circles
- Circumference formula: ( C = 2\pi r ) (example: radius = 5, circumference = ( 10\pi ) or 31.416)
- Area formula: ( A = \pi r^2 ) (example: radius = 5, area = ( 25\pi ) or 78.54)
- Diameter formula: ( d = 2r ) (example: radius = 5, diameter = 10)
- Chord: a segment joining two points on a circle that does not pass through the center
Squares
- Area formula: ( A = s^2 ) (example: side = 8, area = 64)
- Perimeter formula: ( P = 4s ) (example: side = 8, perimeter = 32)
- To find side from area: ( s = \sqrt{\text{area}} )
- Example: area = 36, side = 6, perimeter = 24
Rectangles
- Area formula: ( A = l \times w ) (example: length = 10, width = 5, area = 50)
- Perimeter formula: ( P = 2l + 2w ) (example: length = 10, width = 5, perimeter = 30)
- Given area and one side, find the other: ( w = \frac{\text{area}}{l} )
- Word problems: set up equations based on descriptions (e.g., length is "three more than twice the width")
Problem-Solving with Rectangles
- For area = 44, length ( l = 3 + 2w ): solve ( (3 + 2w)w = 44 ) for width ( w ) using factoring.
- For perimeter = 26, length ( l = 3 + w ): use ( 2l + 2w = 26 ) and solve for both sides, then area.
Triangles and Pythagorean Theorem
- Right triangle: ( a^2 + b^2 = c^2 ) where ( c ) is the hypotenuse.
- Common special right triangles: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 11-60-61.
- Multiples of special triangles (e.g., 6-8-10 from 3-4-5).
Applications and Tips
- Quickly recognize special triangles to save time on exams.
- Use area and perimeter formulas to solve for missing sides.
- For rectangles in problems, draw diagrams and set up equations from word descriptions.
Key Terms & Definitions
- Circumference — distance around a circle
- Diameter — line through circle center touching both edges (( d = 2r ))
- Radius — distance from the center to the circle's edge
- Chord — segment joining two points on a circle, not through center
- Perimeter — distance around a polygon
- Area — space inside a two-dimensional shape
- Pythagorean theorem — relationship in right triangles: ( a^2 + b^2 = c^2 )
- Hypotenuse — longest side of a right triangle, opposite the right angle
Action Items / Next Steps
- Practice solving geometry problems with given formulas.
- Memorize special right triangle sets for quick identification.
- Review and solve word problems using rectangles and squares.