Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
History of trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Incircle and excircles of a triangle wikipedia , lookup
Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ 3.1 Duplicating Segments and Angles [and Triangles] These notes replace pages 144 – 146 in the book. You can read these pages for extra clarifications. Instructions for making geometric figures: You can sketch a figure without using geometry tools. You need to mark the diagram with equal segments, equal angles and parallel lines (or parts of lines) or label the measures of the parts to indicate more accurate measures. You can draw a figure using measuring tools, such as a protractor and a ruler. Make a drawing when it is important for lengths and angle measures to be fairly precise. Mark the measures in the diagrams. You can construct a figure using only a compass and straightedge. When you make a construction, do not use your measuring tools. You MUST show your “arc marks” to show your work!! Duplicate will mean to make an exact copy. Construct a duplicate line segment. Construct an Equilateral Triangle. Page 145 Investigation 1: Construct AB ≅ CD . Stage 1: Draw a ray longer than AB label endpoint C. Stage 2: With compass, measure AB and make an arc. Page 147 #8. Construct equilateral triangle Stage 3: Put point of compass on point C and make an arc. Label the intersection D. sides all equal to AB. Stage 1: Construct EQ ≅ AB . Stage 2: With compass equal to AB. Swing an arc with center E. Stage 3: With compass equal to AB. Swing an arc with center Q. Mark the intersection of the two arcs U. Stage 4: Construct sides EU and QU . A B C A D ∆ EQU , with B U Duplicate a triangle. SSS Method. ∆ TRI ≅ ∆ ABC . Stage 1: Construct TR ≅ AB . Page 147 #7. Construct Stage 2: Use compass to measure TI. Swing an arc with center A. Stage 3: Use compass to measure RI. Swing an arc with center B. Mark the intersection of the two arcs C. Stage 4: Construct sides AC and BC . I E R T Q Can also be used to construct an equiangular triangle, and to construct a 60° angle. C A S. Stirling B Page 1 of 7 Ch 3 Note Sheets S15 KEY LEVEL 2 3.2 Constructing Perpendicular Bisectors Name _________________________ Construct a Perpendicular Bisector A segment bisector is a line, ray, or segment that passes through the midpoint of the segment. A perpendicular bisector of a segment is a line (or part of a line) that passes through the midpoint of a segment and is perpendicular to the segment. A segment has an infinite number of bisectors, but in a plane it has only one perpendicular bisector. Page 150 Investigation 2 Construct CD , the perpendicular bisector of AB . Stage 1: Set compass to a radius longer than 1 AB . 2 Stage 2: With A as center, make an arc above and below AB . Stage 3: With B as center, make an arc above and below AB . Label the intersections C and D. Stage 4: Construct CD . Label intersection M. A B Complete Investigation 1 page 149-150. Perpendicular Bisector Conjecture (P 149 Inv 1): If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints. What is point M? the midpoint Converse of the Perpendicular Bisector Conjecture: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Given that M is the midpoint of AB : E B C Name segment bisectors: CD , ME , CD , ME , etc… Name perpendicular bisectors: M CD , CD , CM , etc… A Since point D is on the perpendicular bisector of D AB , AD = DB. Make a point G that is equidistant from A and B. Where is it? On the perpendicular bisector. Is S. Stirling AB a bisector? A perpendicular bisector? Neither Page 2 of 7 Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ 3.3 Constructing Perpendiculars to a Line Read the top of page 154 in the book and the top of page 156 (1st paragraph). Shortest Distance Conjecture The shortest distance from a point to a line is measured along the perpendicular segment from the point to the line. Draw a Perpendicular Line to a line from a point NOT on the line. Stage 1: Place your protractor on line j forming a 90º angle and slide it until the perpendicular ray goes through P. Stage 2: Draw the perpendicular through P. Label intersection Q. The distance from a point to a line is the length of the perpendicular segment from the point to the line. A R M Q j P B S P What is the shortest distance from P to AB ? MP Is PQ the shortest distance from P to line j? Yes Measure the distance from the mall, point M, to each of the major highways that surround the mall. Use the scale 1 cm = 1 mile. What is the shortest distance from the mall to each route? In other words, find the distance from point M to each line. route 2 route 1 M route 3 S. Stirling Page 3 of 7 Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ Read the rest of page 156 top. The altitude of a triangle is a perpendicular segment from a vertex of a triangle to the line containing the opposite side. Draw all Altitudes [in an acute triangle]. The length of this segment is the height of the triangle. A triangle has three sides, so it has three different altitudes and it has three different heights! In the figures, use a colored pencil to indicate the heights, draw the altitudes, from each side of the triangle. Use a note card or protractor to help you! You may also want to turn the triangle around so that the segment you are drawing the altitude to is parallel to you. Draw all Altitudes [in a right triangle]. Where are all of the altitudes located? All inside the triangle. Draw all Altitudes [in an obtuse triangle]. You will need to extend the sides BO and TO in order to draw the altitudes!! Where are all of the altitudes located? Where are all of the altitudes located? One inside and 2 on the sides of the triangle (the legs). One inside and 2 outside the triangle. As you can see, an altitude can be inside or outside the triangle, or it can be one of the triangle’s sides. It all depends on the shape of the triangle. T T A L A R A L R Acute ∆ TRA with altitude AL . AL is the height from base RT . S. Stirling Right ∆ TRA with altitude AT . AT is the height from base RT . R Obtuse ∆ TRA with altitude AL is the height from base Page 4 of 7 T AL RT Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ 3.4 Constructing Angle Bisectors Read the top of page 159 in the book. Complete Investigation 1 page 159, need tracing paper. P Angle Bisector Conjecture If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. If A is on angle bisector QA , then XA = AY. X A Q or A is equidistant from QP and QR . Y R Does every angle have a bisector? Yes Is it possible for an angle to have more than one bisector? No, only one. Is the converse true? Converse of Angle Bisector Conjecture If a point is equidistant from the sides of the angle, then it is on the bisector of an angle. To show that the converse works, find a point equidistant from the sides of angle BC and a point F on BA that is the same distance from B. So BE = BF . ∠ ABC : 1. Place a point E on A 2. Draw perpendiculars through points E and F. Label the intersection X. Is X equidistant from the sides of the angle? How do you know? Yes. The distance from a point to a line is measured along the perpendicular from the point to the line. Is XE = XF B C ? YES BX . Is it the bisector of ∠ ABC ? YES Measure ∠ CBX and ∠ XBA to confirm. 21º Draw S. Stirling Page 5 of 7 Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ 3.5 Drawing Parallel Lines What properties do you have that can guarantee you parallel lines? If the alternate interior angles are equal, or the corresponding angles are equal, or the same-side interior angles are supplementary, then the lines are parallel. 1. Draw a line parallel to n through P using alternate interior angles. Label what you measured and state the property you used. 2. Draw a line parallel to n through P using corresponding angles. Label what you measured and state the property you used. n n P P 3. Draw a line parallel to n through P using same-side interior angles. Label what you measured and state the property you used. n 107 P 73 S. Stirling Page 6 of 7 Ch 3 Note Sheets S15 KEY LEVEL 2 Name _________________________ Misc. EXERCISES Problem Solving: A. Three towns’ fire stations are shown on the map below. They are planning to join each with straight access roads and then need to locate a central communication tower that is equidistant from the three roads. Find the location of the communication tower and explain to the planners why you know that it is the correct location. The three roads form a triangle. Create the angle bisectors to find a point equidistant from the sides of the triangle, so the tower will be equidistant from the 3 roads. A point on the bisector of an angle is equidistant from the sides of the angle. B. Three towns’ fire stations are shown on the map below. They are planning to consolidate their resources and need to locate a central communication center that is equidistant from the three fire stations. Find the location of the communication center and explain to the planners why you know that it is the correct location. The three roads form a triangle. Create the perpendicular bisectors of the sides to find points equidistant from the endpoints. The point is equidistant from the vertices of the triangle, so the tower will be equidistant from the 3 fire stations. A point on the bisector of a segment is equidistant from the endpoints of the segment. S. Stirling Page 7 of 7