Geometry CP Unit 1 Notes
1.1 The Building Blocks of Geometry The three most basic figures of geometry are:
Points Shown as dots. No size. Named by capital letters. Are collinear if a single line can contain them all. (i.e. line) Are coplanar if a single plane can contain them all (i.e. any three points)
Coplanar v. Collinear Can three collinear points be coplanar? Yes! Can three coplanar points be collinear? No; this is because they might not all necessarily be on the same line.
Lines No thickness. Perfectly straight. Extend foreeeeeeeeeeeeevvvvvvvvvvvvvvvvveeeeeeeeeeeeeeerrrrrrrrrr (in both directions).
Planes They don t fly. Extend infinitely in all directions along a flat surface.
Segments Part of a line that begins at one point and ends at another. Points are called endpoints. This is also known as Postulate 1.1.1
Rays Not from Tampa Bay Part of line that starts at a point and extends infinitely in one direction. Point is called the endpoint of the ray. Known as Postulate 1.1.2
Angles Formed by two rays with a common endpoint. Endpoint is called vertex of the angle Rays make the sides of the angle Angles divide a plane into the interior and the exterior of the angle. Also known as postulate 1.1.3
Intersections Not just limited to streets! When geometric figures have one or more points in common, they are said to intersect. The set of points in common are called their intersection.
So what the heck is a postulate? Throughout the lesson, you ve seen that segments, ray, and angles represent postulate 1.1.1, 1.1.2., and 1.1.3 respectively. A postulate is a fundamental geometry idea. Postulates are statements that are accepted as true without proof. Also known as axioms.
More postulates (page 12 in text) 1.1.4 the intersection of two lines is a point. 1.1.5 the intersection of two planes is a line. 1.1.6 through any two points there is exactly one line. 1.1.7 through any three noncollinear points there is exactly one plane. 1.1.8 if two points are in a plane, then the line containing them is in the plane.
1.2 Measuring Length To determine the length of any segment, we need to use a number line. Number line line that corresponds with real numbers. See the smiley face?
Coordinates Where was the smiley face? That s right; at -2. The smiley face represents a coordinate. Coordinate point on number line that represents a real number.
Postulate 1.2.1 Length Refers to the distance from one point to another. A B Distance in this case goes from either point B to point A or point A to point B. Length can be shown as a- b or b a Substitute in the values in each absolute value equation: -2 4 and 4 (-2) The answer to both problems is 6. Therefore, segment AB is 6 units.
Congruency Congruent figures are the same size and shape. If two lines are congruent, you can take one line and put it on top of the other and they will line up perfectly. Symbol for congruence:
Postulate 1.2.2 Segment Congruence Postulate If two segments have the same length, then the segments are congruent. If two segments are congruent, then they have the same length.
Postulate 1.2.3
In our example, Point Q was in between point P and point R on a line. This means that PQ + QR = PR This is the Segment Addition Postulate.
Additional Geometry Terms Perpendicular Lines two lines that intersect to form a right angle. Parallel Lines two coplanar lines that do not intersect Conjecture statement that you think is true. educated guess
Segment Bisector line that divides a segment into two congruent parts. Midpoint location where a segment is bisected. Perpendicular Bisector line that bisects a segment at exactly 90 degrees. Angle Bisector line or ray that divides an angle into two congruent angles.
Inscribed circle a circle inside a polygon, touching the polygon at its number of points Circumscribed circle circle outside a polygon, contains all vertices of that polygon Center of an inscribed circle is known as the incenter of a triangle. Center of a circumscribed circle is known as the circumcenter of a triangle.
1.6 Motion in Geometry In your groups, huddle up, discuss, and be ready to answer the following questions: What happens if you have an object on your desk, you pick it up, and you put it on the desk behind you? What happens to the minute hand of a clock if you would stare at it long enough (say, about an hour?) What happens if you look into a mirror?
In terms of geometry, all of these questions in the team huddle are examples of motion. Notice that in all of these instances, we did not change size or shape. This is called rigid transformation. In geometry, there are three examples of motion.
Translation (Slide) In translation, every point of a figure moves in a straight line. All points move the same distance. All points move in the same direction. The paths are parallel.
Rotation (Turn) In rotation, every point of a figure moves around a given point known as the center of rotation. A All points move along the same angle measure. In this example, Triangle A is turning?
Reflection (Flip) When you think of a reflection, what do you think of? A reflection (in terms of math) sees every point in a geometric figure flipped.
1.7 Motion in the coordinate plane Remember that when plotting points, (0, 0) is the origin. When plotting points, the x-value is the distance left or right from the origin. Likewise, the y-value is the distance up or down from the x-value. So what are the ordered pairs here?
(1, 7) and (5, 9) So how do we get from point Q to point S? We went to the right 4 and up 2. This is called a transformation. We can use transformation notation to show this.
Transformation Notation T (x, y) = (x + 4, y + 2) This means that from the first point plotted, we went to the right 4 (+ 4) and up 2 (+ 2). T (x, y) = (x 4, y 2) This means that from the first point plotted, we went to the left (-4) and down 2 (-2)
Team Huddle What would the transformation be for these points? From point T to point U? From point S to point R?
Reflections across the x-axis Remember that a reflection is when points are flipped across the line. In this example, the point (2, 3) when flipped across the x-axis leads to the point (2, -3) In other words, when a point is reflected across the x-axis, the y value becomes negative. M (x, y) = (x, -y)
Reflections across the y-axis Remember that a reflection is when points are flipped across the line. In this example, the point (3, 4) when flipped across the y-axis leads to the point (-3, 4) In other words, when a point is reflected across the y-axis, the x value becomes negative. M (x, y) = (-x, y)
Team Huddle How would the point (-3, 5) be reflected across the y-axis? Across the x- axis?
180 Rotations about the Origin In order to figure out a 180 rotation, basically you take your x and y values and multiply them by - 1. If you look at the example, the point (4,2) (shown in blue) was plotted. When you multiply both values by - 1, you get (-4, -2) which is the red point shown to the right.
Team Huddle How would the points (-1, -2), (4, -2), and (-1, 3) be rotated 180 about the origin?
3.8 Analyzing Polygons with Coordinates Buzzwords: Polygon Slope Parallel Perpendicular Mid Reciprocal In your groups, conduct a team huddle and determine what your group already knows about these terms.
This is a coordinate plane. To plot a point, we first have to go left or right (this is the x ). Then we go up or down (this is the y )
What would this point be?
Slope As you can see, we can create triangles by plotting points on the coordinate plane. Using this triangle, we can also find slope.
Slope is rise run Rise is the change in y (or the distance up and down) Run is the change in x (or the distance left to right) What would this slope be?
We can also find the slope of a line if we are given two points. To do this, we use the formula y 2 y 1 x 2 x 1
Example Given the points (5, 3) and (8, 4), find the slope of a line.
Team Huddle In a team huddle, find the slope of a line given the following points. 1. (-3, 5) and (-3, 6) 2. (2, 1) and (4, 1)
Parallel Lines Theorem Two nonvertical lines are parallel if and only if they have the same slope. Any two vertical or horizontal lines are parallel.
Perpendicular Lines Theorem Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Any vertical line is perpendicular to any horizontal line.
Another way to tell if two lines are perpendicular is to look at their slopes. If their slopes are negative reciprocals, then they are perpendicular. Example:
In looking at any shape, we can use slope to determine what type of shape it is. To do this, we have to find the slope of each line. How many different lines can we find the slope for in this figure?
Midpoint Formula If we have two points on a coordinate plane, we can figure out the midpoint (point exactly in between the two points) X 1 + x 2 2 y 1 + y 2 2 In other words, add the x s and divide by 2; add the y s and divide by 2.
Example Given the points (2, 4) and (12, 8), what would the midpoint be?
Team Huddle Given the points (-2, 5) and (16, -1), what would the midpoint be? Given the points (18, 3) and (5, 8), what would the midpoint be?
5.6 The Distance Formula In a team huddle, take 60-90 seconds and discuss what you can remember about coordinate planes.
Distance Formula In plotting two points on a coordinate plane, we can figure out the distance between the points by using the distance formula. Distance Formula: d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2
Example Find the distance between the points (2,7) and (5,3).
Example A waterfall in a park is located 6 miles east and 3 miles north of the entrance. There is a picnic site 1 mile east and 2 miles north of the entrance. How long is a trail that goes directly from the waterfall to the picnic site?