Chapter 5 unit f 001
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This document contains a geometry lesson on perpendicular bisectors and angle bisectors. It begins with definitions of perpendicular bisectors and angle bisectors. It then provides examples of using theorems about perpendicular bisectors and angle bisectors to find unknown measures in geometric figures. It also gives examples of writing equations of perpendicular bisectors and angle bisectors in the coordinate plane. The document aims to prove and apply theorems about perpendicular bisectors and angle bisectors.
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Chapter 5 unit f 001
This document provides definitions, examples, and practice problems related to perpendicular bisectors and angle bisectors. It begins by defining perpendicular bisectors as the locus of points equidistant from the endpoints of a segment. Angle bisectors are defined as the locus of points equidistant from the sides of an angle. Examples show applying theorems about perpendicular and angle bisectors to find missing measures. The document concludes with an example writing an equation for a perpendicular bisector in point-slope form.
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1) Hyperbolas are defined as sets of points where the difference between the distances to two fixed points (foci) is a constant. They can be graphed using the standard form equation.
2) Hyperbolas have two branches, two axes of symmetry, vertices, co-vertices, and asymptotes. The standard form equation depends on whether the transverse axis is horizontal or vertical.
3) Examples show how to write the standard form equation, find vertices/co-vertices/asymptotes, and graph hyperbolas. Parameters like the center, foci and axes can change the graph of the hyperbola.
mathematic project .pdf
Lagrange's Mean Value Theorem, also known as the Mean Value Theorem (MVT), is a fundamental result in calculus that describes the relationship between the slope of a tangent to a function's graph and the average rate of change of the function over an interval. It is a crucial tool in analyzing the behavior of functions and has wide-ranging applications in various areas of mathematics and science. The Mean Value Theorem states that if a function f satisfies the following conditions: 1. Establish inequalities: By comparing the slope of the tangent to the average rate of change, the Mean Value Theorem can be used to establish inequalities involving function values.
2. Prove Rolle's Theorem: Rolle's Theorem is a special case of the Mean Value Theorem that applies to functions that have zero values at the endpoints of an interval. 3. Analyze Rolle's Theorem: The Mean Value Theorem can be used to analyze the conditions for Rolle's Theorem and understand the geometric implications of the theorem.
Coordinate geometry
The document discusses various concepts in coordinate geometry including:
(1) Calculating the gradient of a straight line from its equation in slope-intercept form or given two points on the line. Parallel and perpendicular lines have related gradients.
(2) Finding the midpoint and length of a line segment given two points.
(3) Determining the equation of a straight line given its gradient and a point, or given two points, or from its graph.
(4) Interpreting the x- and y-intercepts from the equation of a straight line.
5-2Bisectors of Triangles.ppsx
This document discusses properties of bisectors of triangles, including perpendicular bisectors and angle bisectors. It provides examples and explanations of key concepts such as: the circumcenter theorem stating that the perpendicular bisectors of a triangle are concurrent at the circumcenter, which is equidistant from the triangle's vertices; the incenter theorem stating that the angle bisectors are concurrent at the incenter, which is equidistant from the triangle's sides; and applications of using bisectors to find equidistant points within triangles.
COORDINATE GEOMETRY
Coordinate geometry is the study of representing geometric figures on two- or three-dimensional planes. Some key concepts in coordinate geometry include:
- Finding the distance between two points using the Pythagorean theorem.
- Finding the midpoint of a line segment by taking the average of the x- and y-coordinates of the two endpoints.
- Dividing a line segment into parts externally or internally based on a given ratio of distances.
- Calculating the gradient or slope of a line as the ratio of vertical to horizontal change between two points on the line.
- Determining whether three points are collinear by checking if they have the same slope.
Gch1 l3
This document provides instruction on measuring and constructing angles:
- It defines angles and how to name them using vertex points.
- Methods for measuring angles using a protractor or transit are described. Angles can be classified as acute, right, or obtuse based on their measure.
- The angle addition and bisector properties are explained. Examples show using these concepts to find unknown angle measures.
- Exercises provide practice measuring, naming, classifying, and finding angles based on given information.
Geometry unit 9.4
This document discusses compositions of isometries such as translations, reflections, and rotations. It provides examples of drawing the results of compositions of isometries by performing one transformation followed by another. It also describes how certain compositions, like a reflection followed by a parallel reflection, are equivalent to a single transformation like a translation. The document aims to help students understand and apply theorems about compositions of isometries.
Geometry unit 9.4
This document discusses compositions of isometries such as translations, reflections, and rotations. It provides examples of drawing the results of compositions of isometries by performing one transformation followed by another. The key points are that a composition of transformations results in an image congruent to the original, and certain compositions are equivalent to single transformations like translations. Glide reflections, which are compositions of translations and reflections, are also introduced.
Lecture 2b Parabolas.pdf
This document provides an overview of parabolas including:
(1) The definition of a parabola as the locus of points equidistant from a fixed point (the focus) and fixed line (the directrix)
(2) The key parts of a parabola including the vertex, axis of symmetry, focus, directrix, and latus rectum
(3) How to write the standard and general forms of the equation of a parabola and how to convert between the two forms
(4) The steps to graph a parabola given its equation in vertex form
Lecture 4.1 4.2
The document discusses classifying and finding angles of triangles. It defines different types of triangles based on sides or angles, such as equilateral, isosceles, scalene, acute, obtuse, right. The triangle sum theorem states the sum of interior angles is 180 degrees. The exterior angle theorem relates exterior to interior angles. Examples show using these theorems to find missing angle measures in various triangles.
Module 2 geometric relations
This module introduces geometric relationships between lines and angles. It discusses parallel and perpendicular lines, and defines perpendicular bisectors of line segments. It also covers exterior angles of triangles and triangle inequalities involving side lengths and angle measures. Key concepts taught include the perpendicular bisector theorem, exterior angle theorem, triangle inequality theorem, and the Pythagorean theorem for right triangles. Students are expected to learn to identify and apply properties of parallel, perpendicular and intersecting lines, and solve problems involving triangle inequalities.
Perpendicular lines, gradients, IB SL Mathematics
Here are the steps to solve these problems:
1. Find the slopes of the two lines:
m1 = (8-2)/(5--2) = 6/3 = 2 (slope of r)
m2 = (7-0)/(-8--2) = 7/-6 = -1 (slope of s)
The slopes are negative reciprocals, so r ⊥ s.
2. The slopes are m1 = 2 and m2 = -1/2. Since m1 × m2 = -1, the lines are perpendicular.
3. The given line has slope 3. The perpendicular line will have slope -1/3. Plug into the point-slope form
1.2 deflection of statically indeterminate beams by moment area method
This document discusses elastic beam theory and how it relates to the bending of beams. It contains the following key points:
1) Elastic beam theory assumes the beam bends into a smooth curve such that cross-sections remain plane and perpendicular to the neutral axis. The radius of curvature is defined as the distance from the center of curvature to the beam.
2) Hooke's law and the flexure formula can be used to relate the radius of curvature to the internal moment and beam properties. Their product is called the flexural rigidity.
3) The moment-area theorems relate the slope and displacement of the beam to the area under the bending moment diagram divided by the flexural rigidity (M/
Stolyarevska_IES
The document discusses using the GeoGebra mathematical software package in mathematics teaching. It presents 14 propositions about ellipses, hyperbolas, and parabolas. Students used GeoGebra's interactive features to construct the curves as loci of points and solve problems of varying complexity based on the propositions. This reinforced their understanding of concepts from the "Second-Order Curves" course material from sources like textbooks and papers.
Ce 255 handout
- The document describes the conjugate beam method for analyzing beams with varying rigidities.
- The method involves drawing an imaginary conjugate beam that is loaded based on the bending moments of the real beam.
- The slope and deflection of points on the real beam can then be determined from the shear and bending moment of the corresponding points on the conjugate beam.
- An example problem is worked out in detail to demonstrate calculating the slope and deflection at the end of a cantilever beam with varying moment of inertia along its length using the conjugate beam method.
Cirlces day 1 and day 2 together
This document contains examples and explanations about lines and circles. It defines key terms like chord, secant, tangent, diameter, and radius. It then provides examples of identifying these features when they intersect circles. Subsequent examples show finding radii, points of tangency, and writing equations of tangent lines. Other examples demonstrate using properties of tangents to solve problems and find measures of arcs and angles related to circles.
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2) Students are asked to work with a partner using devices and packets to investigate triangle properties like perpendicular bisectors, angle bisectors, midsegments, and medians using geometry software.
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The document provides instructions and diagrams for 4 math problems involving angles and perpendicular bisectors. It aims to review skills around finding unknown angles and distances given information about perpendicular or angle bisectors. The final section models explaining geometric proofs through stating reasons and using theorems such as vertical angles, alternate interior angles, and angle-angle-side.
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Chapter 5 unit f 001
- 1. GT Geom. Drill 12/13/13 Define each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and slope of the segment (2, 8) and (–4, 6).
- 2. Objectives Prove and apply theorems about perpendicular bisectors. Prove and apply theorems about angle bisectors.
- 4. When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.
- 7. Based on these theorems, an angle bisector can be defined as the locus of all points in the interior of the angle that are equidistant from the sides of the angle.
- 8. A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.
- 9. Example 1A: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. MN MN = LN Bisector Thm. MN = 2.6 Substitute 2.6 for LN.
- 10. Example 1B: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. BC Since AB = AC and , is the perpendicular bisector of by the Converse of the Perpendicular Bisector Theorem. BC = 2CD Def. of seg. bisector. BC = 2(12) = 24 Substitute 12 for CD.
- 11. Example 1C: Applying the Perpendicular Bisector Theorem and Its Converse Find each measure. TU TU = UV Bisector Thm. 3x + 9 = 7x – 17 Substitute the given values. 9 = 4x – 17 Subtract 3x from both sides. 26 = 4x 6.5 = x Add 17 to both sides. Divide both sides by 4. So TU = 3(6.5) + 9 = 28.5.
- 12. Remember that the distance between a point and a line is the length of the perpendicular segment from the point to the line.
- 13. Example 2A: Applying the Angle Bisector Theorem Find the measure. BC BC = DC Bisector Thm. BC = 7.2 Substitute 7.2 for DC.
- 14. Example 2B: Applying the Angle Bisector Theorem Find the measure. mEFH, given that mEFG = 50°. Since EH = GH, and , bisects EFG by the Converse of the Angle Bisector Theorem. Def. of bisector Substitute 50° for mEFG.
- 15. Example 2C: Applying the Angle Bisector Theorem Find mMKL. Since, JM = LM, and , bisects JKL by the Converse of the Angle Bisector Theorem. mMKL = mJKM Def. of bisector 3a + 20 = 2a + 26 a + 20 = 26 a=6 Substitute the given values. Subtract 2a from both sides. Subtract 20 from both sides. So mMKL = [2(6) + 26]° = 38°
- 16. Example 4: Writing Equations of Bisectors in the Coordinate Plane Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1). Step 1 Graph . The perpendicular bisector of is perpendicular to at its midpoint.
- 17. Example 4 Continued Step 2 Find the midpoint of . Midpoint formula. mdpt. of =
- 18. Example 4 Continued Step 3 Find the slope of the perpendicular bisector. Slope formula. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is
- 19. Example 4 Continued Step 4 Use point-slope form to write an equation. The perpendicular bisector of has slope and passes through (8, –2). y – y1 = m(x – x1) Point-slope form Substitute –2 for y1, for x1. for m, and 8
- 21. Check It Out! Example 4 Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4). Step 1 Graph PQ. The perpendicular bisector of is perpendicular to at its midpoint.
- 22. Check It Out! Example 4 Continued Step 2 Find the midpoint of PQ. Midpoint formula.
- 23. Check It Out! Example 4 Continued Step 3 Find the slope of the perpendicular bisector. Slope formula. Since the slopes of perpendicular lines are opposite reciprocals, the slope of the perpendicular bisector is .
- 24. Check It Out! Example 4 Continued Step 4 Use point-slope form to write an equation. The perpendicular bisector of PQ has slope passes through (3, –1). y – y1 = m(x – x1) Point-slope form Substitute. and
- 25. Use the diagram for Items 1–2. 1. Given that mABD = 16°, find mABC. 32° 2. Given that mABD = (2x + 12)° and mCBD = (6x – 18)°, find mABC. 54° Use the diagram for Items 3–4. 3. Given that FH is the perpendicular bisector of EG, EF = 4y – 3, and FG = 6y – 37, find FG. 65 4. Given that EF = 10.6, EH = 4.3, and FG = 8.6 10.6, find EG.
- 26. Assignment #48? • Handout from book • P304-305 (myhrw.com)#12- 32, for 30 and 31 two column proof • Here is the picture blown up for 23-28