Example Lines through Noncollinear Points
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The document considers 5 non-collinear points (A, B, C, D, E) and calculates the number of lines that can be drawn between two points. It determines that there are 4 lines through point A, 3 additional lines through point B, 2 more lines through point C, and 1 final line through point D/E, for a total of 4 + 3 + 2 + 1 = 10 lines.
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- A plane is a flat, two-dimensional surface that can be named using points or a cursive letter
Section 1
The document defines and provides examples of points, lines, and planes in geometry. It states that a point is a location, a line is made up of collinear points with no thickness or width, and a plane is a flat surface made up of coplanar points that extends infinitely in all directions. It also notes that coplanar points all lie in the same plane, while collinear points all lie on the same line.
1 3 points, lines, planes
This document defines key geometric concepts including points, lines, planes and their relationships. It explains that points have no size, lines extend indefinitely and planes are flat surfaces that extend without limits. It also covers topics like collinear points that lie on the same line, determining if objects are coplanar by lying on the same plane, and the different types of intersections between lines and planes, which can be a point, line or no intersection.
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Converting Metric Units
This document explains how to convert 54.6 meters to kilometers. It shows that to convert meters to kilometers, we move the decimal place three places to the left. Therefore, 54.6 meters converts to 0.0546 kilometers.
Solve Systems using Augmented Matrices
The document solves the system of equations x + 2y = 16 and 2x + y = 11 using augmented matrices. It first writes the system as an augmented matrix and then performs row operations, multiplying rows by constants and adding rows together, to transform the matrix into an identity matrix with the solutions in the right column. The solutions are x = 2 and y = 7.
Solve by Substitution
The document shows the steps to solve the system of equations 5x - y = 5 and 4x - y = 3 by substitution. It involves solving the first equation for y in terms of x, substituting this expression for y into the second equation, solving the resulting equation for x, and then substituting x back into the original equation to solve for y. The solution is the ordered pair (2,5).
Adding and subtracting in other bases
The document shows the step-by-step workings of two base conversions:
1) 302.45 + 402.55 in base 5, which equals 121045
2) 701.48 - 340.78 in base 8, which equals 34058
It breaks down each digit addition and subtraction, borrowing when needed, to get the final answers in the given bases.
Writing Hindu-Arabic Numerals in Expanded Form
This document explains how to write numbers in expanded form. It shows working through writing 472 and 12,357 in expanded form. To write a number in expanded form, you start with the rightmost digit and multiply it by increasing powers of 10 as you move left. So 472 in expanded form is (7 × 101) + (4 × 102) + (2 × 100).
Dividing fractions examples
This document provides step-by-step instructions for dividing fractions. It shows dividing 2/3 by 6/7, which involves inverting the second fraction and multiplying. The resulting fraction is then simplified by factoring out common factors to get 1/3. A second example divides -5/8 by 4/5, again inverting the second fraction and multiplying before giving the final simplified fraction of -25/32.
Multiplying fractions example
The document provides steps for multiplying fractions. It shows:
1) Multiplying 2/3 by 7/16 by finding a common factor of 2 in the denominators and simplifying to get 7/24.
2) Multiplying 1 3/4 by 2 1/2 by first converting to improper fractions, then multiplying numerators and denominators to get 35/8.
3) Converting the final fraction 35/8 to a mixed number of 4 3/8.
Division method example solution
The document provides steps to find the prime factors of 198 using the division method. It is first divided by 2, then the resulting number 99 is divided by 3 to get 33, which is then divided by 3 to get 11. Since 11 is prime, the prime factors of 198 are 2, 3, and 11.
Branching Example
This document outlines the steps to use the branching method to find the prime factors of 315. It explains that 315 is divisible by 5, 3, and 9, so the first step is to divide 315 by 15 to get 21. Then 21 is divided by 3 and 7, the prime factors of 21. The prime factors of 315 are then expressed as the product 3 x 3 x 5 x 7.
Section 3-1: Congruent Triangles
Two triangles are congruent if their corresponding six parts (three angles and three sides) are congruent. Congruence can also be described as one triangle coinciding perfectly with the other after rotation or flipping. In the figure, triangle ABC is congruent to triangle DEF as demonstrated by their corresponding angles and sides being congruent. Congruence of triangles is an equivalence relation that is reflexive, symmetric, and transitive.
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Example Lines through Noncollinear Points
- 1. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE
- 2. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE There are 4 lines through point A and one other point 4
- 3. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE There are 3 additional lines through point B and one additional point 4 + 3
- 4. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE There are 2 additional lines through point C and one additional point 4 + 3 + 2
- 5. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE There is 1 additional line through point C and one additional point 4 + 3 + 2 + 1
- 6. Example: Lines through Non-collinear Points Consider points A, B, C, D, and E no three of which are collinear. How many lines can be drawn through these points such that each line contains two of the points? A B C DE 4 + 3 + 2 + 1 = 10 10 lines