LAB # SEASONAL PATH of the SUN and LATITUDE Hemisphere
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Angle Chasing
Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle. -
Point of Concurrency the Three Perpendicular Bisectors of a Triangle Intersect at a Single Point
3.1 Special Segments and Centers of Classifications of Triangles: Triangles By Side: 1. Equilateral: A triangle with three I CAN... congruent sides. Define and recognize perpendicular bisectors, angle bisectors, medians, 2. Isosceles: A triangle with at least and altitudes. two congruent sides. 3. Scalene: A triangle with three sides Define and recognize points of having different lengths. (no sides are concurrency. congruent) Jul 249:36 AM Jul 249:36 AM Classifications of Triangles: Special Segments and Centers in Triangles By angle A Perpendicular Bisector is a segment or line 1. Acute: A triangle with three acute that passes through the midpoint of a side and angles. is perpendicular to that side. 2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle 4. Equiangular: A triangle with three congruent angles Jul 249:36 AM Jul 249:36 AM Point of Concurrency The three perpendicular bisectors of a triangle intersect at a single point. Two lines intersect at a point. The point of When three or more lines intersect at the concurrency of the same point, it is called a "Point of perpendicular Concurrency." bisectors is called the circumcenter. Jul 249:36 AM Jul 249:36 AM 1 Circumcenter Properties An angle bisector is a segment that divides 1. The circumcenter is an angle into two congruent angles. the center of the circumscribed circle. BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. m∠ABD= m∠DBC Jul 249:36 AM Jul 249:36 AM The three angle bisectors of a triangle Incenter properties intersect at a single point. -
Basic Principles of Celestial Navigation James A
Basic principles of celestial navigation James A. Van Allena) Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242 ͑Received 16 January 2004; accepted 10 June 2004͒ Celestial navigation is a technique for determining one’s geographic position by the observation of identified stars, identified planets, the Sun, and the Moon. This subject has a multitude of refinements which, although valuable to a professional navigator, tend to obscure the basic principles. I describe these principles, give an analytical solution of the classical two-star-sight problem without any dependence on prior knowledge of position, and include several examples. Some approximations and simplifications are made in the interest of clarity. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1778391͔ I. INTRODUCTION longitude ⌳ is between 0° and 360°, although often it is convenient to take the longitude westward of the prime me- Celestial navigation is a technique for determining one’s ridian to be between 0° and Ϫ180°. The longitude of P also geographic position by the observation of identified stars, can be specified by the plane angle in the equatorial plane identified planets, the Sun, and the Moon. Its basic principles whose vertex is at O with one radial line through the point at are a combination of rudimentary astronomical knowledge 1–3 which the meridian through P intersects the equatorial plane and spherical trigonometry. and the other radial line through the point G at which the Anyone who has been on a ship that is remote from any prime meridian intersects the equatorial plane ͑see Fig. -
Approved Student Calendar
2007-2008 Student Calendar July 2007 August 2007 September 2007 SMTWT F S SMTWT F S SMTWT F S 1234567 1234 1 8910111213145678910 11 2 3 45678 15 16 17 18 19 20 21 12 13 14 15 16 17 18 9 10 11 12 13 14 15 22 23 24 25 26 27 28 19 20 21 22 23 24 25 16 17 18 19 20 21 22 23 29 30 31 26 27 28 29 30 31 30 24 25 26 27 28 29 October 2007 November 2007 December 2007 SMTWT F S SMTWT F S SMTWT F S 123456 123 1 7891011 12 134567 89102345678 14 15 16 17 18 19 20 11 12 13 14 15 16 17 9 10 11 12 13 14 15 21 22 23 24 25 26 27 18 19 20 21 22 23 24 16 17 18 19 20 21 22 23 24 28 29 30 31 25 26 27 28 29 30 30 31 25 26 27 28 29 January 2008 February 2008 March 2008 SMTWT F S SMTWT F S SMTWT F S 12345 12 1 67891011123456789 2345678 13 14 15 16 17 18 19 10 11 12 13 14 1516 9 1011121314 15 20 21 22 23 24 25 26 17 18 19 20 21 22 23 16 17 18 19 20 21 22 23 24 27 28 29 30 31 24 25 26 27 28 29 30 31 25 26 27 28 29 April 2008 May 2008 June 2008 SMTWT F S SMTWT F S SMTWT F S 12345 123 1234567 6789 10111245678910891011121314 13 14 15 16 17 18 19 11 12 13 14 15 16 17 15 16 17 18 19 20 21 20 21 22 23 24 25 26 18 19 20 21 22 23 24 22 23 24 25 26 27 28 27 28 29 30 25 26 27 28 29 30 31 29 30 Regular School Day Schools Closed Early Dismissal Parent Conference Report Card Distribution * This calendar was adjusted to accommodate the spring NASCAR race. -
DTA Scoring Form
Daytime Astronomy SHADOWS Score Form Date ____/____/____ Student _______________________________________ Rater _________________ Problem I. (Page 3) Accuracy of results Draw a dot on this map to show where you think Tower C is. Tower C is in Eastern US Tower C is in North Eastern US Tower C is somewhere in between Pennsylvania and Maine Modeling/reasoning/observation How did you figure out where Tower C is? Matched shadow lengths of towers A and B/ pointed flashlight to Equator Tried different locations for tower C/ inferred location of tower C Matched length of shadow C/considered Latitude (distance from Equator) Matched angle of shadow C/ considered Longitude (East-West direction) Problem II. (Page 4) Accuracy of results How does the shadow of Tower A look at 10 AM and 3 PM?... Which direction is the shadow moving? 10 AM shadow points to NNW 10 AM shadow is shorter than noon shadow 3 PM shadow points to NE 3 PM shadow is longer than noon shadow Clockwise motion of shadows Modeling/reasoning/observation How did you figure out what the shadow of Tower A looks like at 10 AM and 3 PM? Earth rotation/ "Sun motion" Sunlight coming from East projects a shadow oriented to West Sunlight coming from West projects a shadow oriented to East Sunlight coming from above us projects a shadow oriented to North Sun shadows are longer in the morning than at noon Morning Sun shadows become shorter and shorter until its noon The shortest Sun shadow is at noon Sun shadows are longer in the afternoon than at noon Afternoon Sun shadows become longer and longer until it gets dark (Over, please) - 1 - 6/95 Problem III. -
Median and Altitude of a Triangle Goal: • to Use Properties of the Medians
Median and Altitude of a Triangle Goal: • To use properties of the medians of a triangle. • To use properties of the altitudes of a triangle. Median of a Triangle Median of a Triangle – a segment whose endpoints are the vertex of a triangle and the midpoint of the opposite side. Vertex Median Median of an Obtuse Triangle A D Point of concurrency “P” or centroid E P C B F Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. A D If P is the centroid of ABC, then AP=2 AF E 3 P C CP=22CE and BP= BD B F 33 Example - Medians of a Triangle P is the centroid of ABC. PF 5 Find AF and AP A D E P C B F 5 Median of an Acute Triangle A Point of concurrency “P” or centroid E D P C B F Median of a Right Triangle A F Point of concurrency “P” or centroid E P C B D The three medians of an obtuse, acute, and a right triangle always meet inside the triangle. Altitude of a Triangle Altitude of a triangle – the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side A altitude C B Altitude of an Acute Triangle A Point of concurrency “P” or orthocenter P C B The point of concurrency called the orthocenter lies inside the triangle. Altitude of a Right Triangle The two legs are the altitudes A The point of concurrency called the orthocenter lies on the triangle. -
The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars
Heavenly Mathematics: The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars Helmer Aslaksen Department of Mathematics National University of Singapore [email protected] www.math.nus.edu.sg/aslaksen/ www.chinesecalendar.net 1 Public Holidays There are 11 public holidays in Singapore. Three of them are secular. 1. New Year’s Day 2. Labour Day 3. National Day The remaining eight cultural, racial or reli- gious holidays consist of two Chinese, two Muslim, two Indian and two Christian. 2 Cultural, Racial or Religious Holidays 1. Chinese New Year and day after 2. Good Friday 3. Vesak Day 4. Deepavali 5. Christmas Day 6. Hari Raya Puasa 7. Hari Raya Haji Listed in order, except for the Muslim hol- idays, which can occur anytime during the year. Christmas Day falls on a fixed date, but all the others move. 3 A Quick Course in Astronomy The Earth revolves counterclockwise around the Sun in an elliptical orbit. The Earth ro- tates counterclockwise around an axis that is tilted 23.5 degrees. March equinox June December solstice solstice September equinox E E N S N S W W June equi Dec June equi Dec sol sol sol sol Beijing Singapore In the northern hemisphere, the day will be longest at the June solstice and shortest at the December solstice. At the two equinoxes day and night will be equally long. The equi- noxes and solstices are called the seasonal markers. 4 The Year The tropical year (or solar year) is the time from one March equinox to the next. The mean value is 365.2422 days. -
The Basques of Lapurdi, Zuberoa, and Lower Navarre Their History and Their Traditions
Center for Basque Studies Basque Classics Series, No. 6 The Basques of Lapurdi, Zuberoa, and Lower Navarre Their History and Their Traditions by Philippe Veyrin Translated by Andrew Brown Center for Basque Studies University of Nevada, Reno Reno, Nevada This book was published with generous financial support obtained by the Association of Friends of the Center for Basque Studies from the Provincial Government of Bizkaia. Basque Classics Series, No. 6 Series Editors: William A. Douglass, Gregorio Monreal, and Pello Salaburu Center for Basque Studies University of Nevada, Reno Reno, Nevada 89557 http://basque.unr.edu Copyright © 2011 by the Center for Basque Studies All rights reserved. Printed in the United States of America Cover and series design © 2011 by Jose Luis Agote Cover illustration: Xiberoko maskaradak (Maskaradak of Zuberoa), drawing by Paul-Adolph Kaufman, 1906 Library of Congress Cataloging-in-Publication Data Veyrin, Philippe, 1900-1962. [Basques de Labourd, de Soule et de Basse Navarre. English] The Basques of Lapurdi, Zuberoa, and Lower Navarre : their history and their traditions / by Philippe Veyrin ; with an introduction by Sandra Ott ; translated by Andrew Brown. p. cm. Translation of: Les Basques, de Labourd, de Soule et de Basse Navarre Includes bibliographical references and index. Summary: “Classic book on the Basques of Iparralde (French Basque Country) originally published in 1942, treating Basque history and culture in the region”--Provided by publisher. ISBN 978-1-877802-99-7 (hardcover) 1. Pays Basque (France)--Description and travel. 2. Pays Basque (France)-- History. I. Title. DC611.B313V513 2011 944’.716--dc22 2011001810 Contents List of Illustrations..................................................... vii Note on Basque Orthography......................................... -
Hogmanay Rituals: Scotland’S New Year’S Eve Celebrations
Accepted Manuscript of a book chapter published by Routledge Suggested citation: Frew, E. & Mair. J. (2014). Hogmanay Rituals: Scotland’s New Year’s Eve Celebrations. In Frost, W. & Laing, J. (Eds) Rituals and traditional events in the modern world. Routledge Hogmanay Rituals: Scotland’s New Year’s Eve Celebrations Elspeth A. Frew La Trobe University Judith Mair Department of Management Monash University ABSTRACT Hogmanay, which is the name given to New Year’s Eve in Scotland, is a long-standing festival with roots going far back into pagan times. However, such festivals are losing their traditions and are becoming almost generic public celebrations devoid of the original rites and rituals that originally made them unique. Using the framework of Falassi’s (1987) festival rites and rituals, this chapter utilises a duoethnographic approach to examine Hogmanay traditions in contemporary Scotland and the extent to which these have been transferred to another country. The chapter reflects on the traditions which have survived and those that have been consigned to history. Hogmanay and Paganism: Scotland’s New Year’s Eve Celebrations INTRODUCTION New Year’s Eve (or Hogmanay as it is known in Scotland) is celebrated in many countries around the world, and often takes the form of a public celebration with fireworks, music and a carnival atmosphere. However, Hogmanay itself is a long-standing festival in Scotland with roots going far back into pagan times. Some of the rites and rituals associated with Hogmanay are centuries old, and the tradition of celebrating New Year Eve (as Hogmanay) on a grander scale than Christmas has been a part of Scottish life for many hundreds of years. -
Sunrise and Sunset Azimuths in the Planning of Ancient Chinese Towns Amelia Carolina Sparavigna
Sunrise and Sunset Azimuths in the Planning of Ancient Chinese Towns Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Sunrise and Sunset Azimuths in the Planning of Ancient Chinese Towns. International Journal of Sciences, Alkhaer, UK, 2013, 2 (11), pp.52-59. 10.18483/ijSci.334. hal- 02264434 HAL Id: hal-02264434 https://hal.archives-ouvertes.fr/hal-02264434 Submitted on 8 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1Department of Applied Science and Technology, Politecnico di Torino, Italy Abstract: In the planning of some Chinese towns we can see an evident orientation with the cardinal direction north-south. However, other features reveal a possible orientation with the directions of sunrise and sunset on solstices too, as in the case of Shangdu (Xanadu), the summer capital of Kublai Khan. Here we discuss some other examples of a possible solar orientation in the planning of ancient towns. We will analyse the plans of Xi’an, Khanbalik and Dali. Keywords: Satellite Imagery, Orientation, Archaeoastronomy, China 1. Introduction different from a solar orientation with sunrise and Recently we have discussed a possible solar sunset directions. -
Sarah Provancher Jeanne Hilt (502) 439-7138 (502) 614-4122 [email protected] [email protected]
FOR IMMEDIATE RELEASE: Tuesday, June 5, 2018 CONTACT: Sarah Provancher Jeanne Hilt (502) 439-7138 (502) 614-4122 [email protected] [email protected] DOWNTOWN TO SHOWCASE FÊTE DE LA MUSIQUE LOUISVILLE ON JUNE 21 ST Downtown Louisville to celebrate the Summer Solstice with live music and more than 30 street performers for a day-long celebration of French culture Louisville, KY – A little taste of France is coming to Downtown Louisville on the summer solstice, Thursday, June 21st, with Fête de la Musique Louisville (pronounced fet de la myzic). The event, which means “celebration of music” in French, has been taking place in Paris on the summer solstice since 1982. Downtown Louisville’s “celebration of music” is presented by Alliance Francaise de Louisville in conjunction with the Louisville Downtown Partnership (LDP). Music will literally fill the downtown streets all day on the 21st with dozens of free live performances over the lunchtime hour and during a special Happy Hour showcase on the front steps of the Kentucky Center from 5:30-7:30pm. A list of performance venues with scheduled performers from 11am-1pm are: Fourth Street Live! – Classical musicians including The NouLou Chamber Players, 90.5 WUOL Young Musicians and Harin Oh, GFA Youth Guitar Summit Kindred Plaza – Hewn From The Mountain, a local Irish band 400 W. Market Plaza – FrenchAxe, a French band from Cincinnati Old Forester Distillery – A local jug band The salsa band Milenio is the scheduled performance group for the Happy Hour also at Fourth Street Live! from 5 – 7 pm. "In Paris and throughout France, the Fête de la Musique is literally 24 hours filled with music by all types of musicians of all skill levels. -
Regents and Midterm Prep Answers
Name__________________________________!Regents and Mid Term Preparation The Seasons Description Position Description Position March 21st B South Pole-24 Hrs of Dark C June 21st C High Kinetic Energy A December 21st A Low Kinetic Energy C September 23rd D Earth Close to Sun A Northern Hemisphere Winter A Earth Far from Sun C Northern Hemisphere Spring B Southern Hemisphere Spring D North Hemisphere Summer C Southern Hemisphere Fall B Northern Hemisphere Fall D Southern Hemisphere Winter C Greatest Orbital Velocity A South Hemisphere Summer A Least Orbital Velocity C 9 Hrs of Day in NYS A 23 1/2 N-Zenith C 12 Hrs of Day in NYS D,B 0 (Equator)-Zenith D,B 15 Hrs of Day in NYS C 23 1/2 S-Zenith A Winter Solstice A North Pole-24 Hrs Day C Vernal Equinox B South Pole-24 Hrs Day A Autumnal Equinox D North Pole-24 Hrs Dark A Summer Solstice C Name__________________________________!Regents and Mid Term Preparation Sunʼs Path in NYS 1. What direction does the sun rise in summer? _____NE___________________ 2. What direction does the sun rise in winter? _________SE_________________ 3. What direction does the sun rise in fall/spring? ________E_______________ 4. How long is the sun out in fall/spring? ____12________ 5. How long is the sun out in winter? _________9______ 6. How long is the sun out in summer? ________15_______ 7. What direction do you look to see the noon time sun? _____S________ 8. What direction do you look to see polaris? _______N_________ 9. From sunrise to noon, what happens to the length of a shadow? ___SMALLER___ 10.