Geometry Cheat Sheet
1. Basic Concepts
1.1 Points, Lines, and Planes
Point: An exact location in space, represented by a dot and having no dimension.
Line: A straight path extending in both directions with no end, having one dimension (length).
Plane: A flat surface extending in all directions, having two dimensions (length and width).
Example:
Identify points, lines, and planes in a given geometric figure.
- Points: \(A\), \(B\), \(C\)
- Lines: Line \(AB\), Line \(BC\)
- Planes: Plane \(ABC\)
1.2 Angles
Angle: Formed by two rays with a common endpoint called the vertex.
Types of Angles:
- Acute Angle: Less than 90°
- Right Angle: Exactly 90°
- Obtuse Angle: Greater than 90° and less than 180°
- Straight Angle: Exactly 180°
Example:
Classify the following angles:
- 45°: Acute Angle
- 90°: Right Angle
- 135°: Obtuse Angle
- 180°: Straight Angle
2. Triangles
2.1 Types of Triangles
Triangles can be classified by their sides and angles:
By Sides:
- Equilateral Triangle: All sides are equal.
- Isosceles Triangle: Two sides are equal.
- Scalene Triangle: All sides are different.
By Angles:
- Acute Triangle: All angles are less than 90°.
- Right Triangle: One angle is 90°.
- Obtuse Triangle: One angle is greater than 90°.
Example:
Identify the type of triangle based on the given side lengths or angles.
- Side lengths 3, 3, 3: Equilateral Triangle
- Side lengths 4, 4, 6: Isosceles Triangle
- Angles 60°, 60°, 60°: Equilateral Triangle
- Angles 90°, 45°, 45°: Right Triangle
2.2 Triangle Properties
Sum of Angles: The sum of the interior angles of a triangle is always 180°.
Pythagorean Theorem: In a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
Congruence: Triangles are congruent if they have the same size and shape, which can be proved by the following criteria:
- SSS: Side-Side-Side
- SAS: Side-Angle-Side
- ASA: Angle-Side-Angle
- AAS: Angle-Angle-Side
- HL: Hypotenuse-Leg (right triangles)
Example:
Use the Pythagorean Theorem to find the missing side of a right triangle with legs 3 and 4.
- Use the formula: \(a^2 + b^2 = c^2\)
- Substitute the given values: \(3^2 + 4^2 = c^2\)
- Simplify: \(9 + 16 = c^2\)
- Result: \(c = \sqrt{25} = 5\)
3. Quadrilaterals
3.1 Types of Quadrilaterals
Quadrilaterals are four-sided polygons with various properties:
- Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal.
- Rectangle: A parallelogram with four right angles.
- Square: A rectangle with all sides equal.
- Rhombus: A parallelogram with all sides equal.
- Trapezoid: At least one pair of opposite sides are parallel.
Example:
Classify the quadrilateral based on the given properties.
- Four equal sides, opposite angles are equal: Rhombus
- Four right angles, opposite sides are equal: Rectangle
- Only one pair of opposite sides are parallel: Trapezoid
3.2 Properties of Quadrilaterals
Area and Perimeter:
- Rectangle: Area = \(l \times w\), Perimeter = \(2(l + w)\)
- Square: Area = \(s^2\), Perimeter = \(4s\)
- Parallelogram: Area = \(b \times h\), Perimeter = \(2(a + b)\)
- Trapezoid: Area = \(\frac{1}{2}(b_1 + b_2) \times h\), Perimeter = sum of all sides
- Rhombus: Area = \(\frac{1}{2}(d_1 \times d_2)\), Perimeter = \(4s\)
Example:
Find the area and perimeter of a rectangle with length 8 and width 5.
- Area = \(8 \times 5 = 40\)
- Perimeter = \(2(8 + 5) = 26\)
- Result: Area = 40, Perimeter = 26
4. Circles
4.1 Circle Properties
A circle is a set of all points in a plane equidistant from a fixed point called the center. The distance from the center to any point on the circle is the radius.
Key Concepts:
- Circumference: The distance around the circle. \(C = 2\pi r\) or \(C = \pi d\).
- Area: The space enclosed by the circle. \(A = \pi r^2\).
- Arc Length: The distance along the arc (a portion of the circumference). \(L = \frac{\theta}{360} \times 2\pi r\), where \(\theta\) is the central angle in degrees.
- Sector Area: The area of a sector (a "slice" of the circle). \(A_{\text{sector}} = \frac{\theta}{360} \times \pi r^2\).
Example:
Find the area and circumference of a circle with radius 7.
- Area = \(\pi \times 7^2 = 49\pi\)
- Circumference = \(2\pi \times 7 = 14\pi\)
- Result: Area = 49\(\pi\), Circumference = 14\(\pi\)
5. Coordinate Geometry
5.1 Distance Formula
The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane can be found using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Example:
Find the distance between the points (3, 4) and (7, 1).
- Substitute the points into the formula: \(d = \sqrt{(7 - 3)^2 + (1 - 4)^2}\)
- Simplify: \(d = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25}\)
- Result: \(d = 5\)
5.2 Midpoint Formula
The midpoint of the segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Example:
Find the midpoint between the points (3, 4) and (7, 1).
- Substitute the points into the formula: \(\text{Midpoint} = \left( \frac{3 + 7}{2}, \frac{4 + 1}{2} \right)\)
- Simplify: \(\text{Midpoint} = (5, 2.5)\)
- Result: Midpoint = (5, 2.5)
5.3 Slope Formula
The slope of a line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Example:
Find the slope of the line passing through the points (3, 4) and (7, 1).
- Substitute the points into the formula: \(m = \frac{1 - 4}{7 - 3} = \frac{-3}{4}\)
- Result: Slope = \(-\frac{3}{4}\)
6. Transformations
6.1 Types of Transformations
Transformations change the position or size of a figure. The main types of transformations are:
- Translation: Slides a figure horizontally, vertically, or both.
- Reflection: Flips a figure over a line (e.g., x-axis, y-axis).
- Rotation: Turns a figure around a point (e.g., origin).
- Dilation: Resizes a figure proportionally from a center point.
Example:
Apply the following transformations to the point (3, 4): Reflect over the x-axis, then translate 2 units down.
- Reflection over the x-axis: The new point is (3, -4).
- Translate 2 units down: The final point is (3, -6).
- Result: The transformed point is (3, -6).
6.2 Similarity and Congruence
Similarity: Figures are similar if they have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are proportional.
Congruence: Figures are congruent if they have the same shape and size. Corresponding angles and sides are equal.
Example:
Determine if the following triangles are similar or congruent based on their side lengths: Triangle A (3, 4, 5) and Triangle B (6, 8, 10).
- Check for similarity: The side lengths are proportional (3/6 = 4/8 = 5/10).
- Check for congruence: The side lengths are not equal, so they are not congruent.
- Result: The triangles are similar but not congruent.
7. Surface Area and Volume
7.1 Surface Area Formulas
Prism: \(SA = 2B + Ph\), where \(B\) is the area of the base, \(P\) is the perimeter of the base, and \(h\) is the height.
Cylinder: \(SA = 2\pi r^2 + 2\pi rh\)
Pyramid: \(SA = B + \frac{1}{2}Pl\), where \(B\) is the area of the base, \(P\) is the perimeter of the base, and \(l\) is the slant height.
Cone: \(SA = \pi r^2 + \pi rl\)
Sphere: \(SA = 4\pi r^2\)
Example:
Find the surface area of a cylinder with radius 3 and height 5.
- Use the formula: \(SA = 2\pi r^2 + 2\pi rh\)
- Substitute the given values: \(SA = 2\pi (3^2) + 2\pi (3)(5)\)
- Simplify: \(SA = 18\pi + 30\pi = 48\pi\)
- Result: Surface Area = 48\(\pi\)
7.2 Volume Formulas
Prism: \(V = Bh\), where \(B\) is the area of the base and \(h\) is the height.
Cylinder: \(V = \pi r^2h\)
Pyramid: \(V = \frac{1}{3}Bh\)
Cone: \(V = \frac{1}{3}\pi r^2h\)
Sphere: \(V = \frac{4}{3}\pi r^3\)
Example:
Find the volume of a cone with radius 3 and height 5.
- Use the formula: \(V = \frac{1}{3}\pi r^2h\)
- Substitute the given values: \(V = \frac{1}{3}\pi (3^2)(5)\)
- Simplify: \(V = \frac{1}{3}\pi (9)(5) = 15\pi\)
- Result: Volume = 15\(\pi\)