Unit 1 Notes Period 9

Day 1-2 Sec 1.1 : Pre-Req Targets: • Slope, Writing the Equation of a line, Vertical and Horizontal lines, parallel and perpendicular lines Graph Given 2 ordered pairs (x1, y1) and (x2, y2) ( x2 , y2 ) ( x1 , y1 ) Run SLOPE Positive Slope Negative Slope Zero Slope Undefined Slope I Fa I 1. Rise Determine the slope between each pair of points. SOLUTION STEPS A (2,4) B (-1,6) Is 2. 2 m 1. 2. 3. 4. Label x1, y1, x2, y2 Substitute each value into the formula Simplify numerator & Denominator Simplify Fraction (If Possible) Determine the slope of lines from the graph. SOLUTION f STEPS le T E 1. 2. 3. 4. 5. Choose two points to use as reference points Count the rise/fall Count the run 𝑅𝐼𝑆𝐸 Write your slope as 𝑚 = 𝑅𝑈𝑁 Simplify Fraction (If Possible) m 1 Writing the Equation of a line Linear equations have multiple ways that they can be written. Each of the different forms has its own strengths. Slope Intercept Form 𝒚 = 𝒎𝒙 + 𝒃 𝒎 = 𝒔𝒍𝒐𝒑𝒆 𝒃 = 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 Strengths of this form: Graphing a line by determining the SLOPE and the Y-INTERCEPT easily. 3. Point Slope Form Standard Form 𝒚−𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏 ) 𝑨𝒙 + 𝑩𝒚 = 𝑪 𝒎 = 𝒔𝒍𝒐𝒑𝒆 𝑨, 𝑩, 𝑪 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕𝒔 (𝒙𝟏 , 𝒚𝟏 ) = 𝒑𝒐𝒊𝒏𝒕 𝒐𝒏 𝒍𝒊𝒏𝒆 Strengths of this form: Easy to Strengths of this form: Easy to find create an equation from the slope and X and Y intercepts. any point on the line. This form comes from the slope Determine the equation of the line. If needed, rewrite the equation in slope-intercept form. “WHAT AM I GIVEN” Slope & Y- intercept 𝒎=− 𝟐 𝑨(𝟎, −𝟓) 𝟑 Slope and Point on Line 𝒎= 𝟏 𝑩(𝟒, −𝟑) 𝟐 Two Points 𝑨(𝟐, 𝟑) 𝑩(−𝟒, −𝟏𝟓) 3 1732 y It.IEIEIIy 3 t2 y y y s 5 z 3 x 3x s 6 3 2 Vertical/Horizontal Lines HORIZONTAL LINE VERTICAL LINE vertical H horizontal 0 slope _O Y 4. y undefined slope equation equation x Write the equation of the line from the given description. a) a horizontal line through (5,0) Ho z y b) a vertical line through (-4,7) VU O x Parallel and Perpendicular Slopes RECALL: Parallel lines are lines that never RECALL: Perpendicular lines are lines that _______________ intersect intersect at a _________________ From Algebra, we know From Algebra, we know Parallel lines have Perpendicular lines have slopes that are the same slope ______________________________ 9Oyrightangle opposite reciprocals _____________________ 1. change 2. flip the the sign over 3 5. Determine if the following lines are parallel, perpendicular or neither. 1 a) 𝑙𝑖𝑛𝑒 1: 𝑦 = − 2 𝑥 + 5 b) 𝑙𝑖𝑛𝑒 2: 𝑦 = 2𝑥 + 9 slopes The lines are (circle one): Parallel 𝑙𝑖𝑛𝑒 1: 𝑦 = − 1𝑥 + 5 𝑙𝑖𝑛𝑒 2: 6𝑥 − 2𝑦 = −4 an slopes oppositereciprocalsThe lines are (circle one): Perpendicular Parallel Neither 3 and l Perpendicular Neither Writing Equations of Parallel and Perpendicular Lines 6. a) Determine the equations for the line described. 3x + 2y = 12 and through the point (0,-3) x Parallel to: 1 11slope b) Izzy y I G24 1m Parallel to: YI y 312 6 y = 4 x − 2 and through the point (-12,4) TX y 3 4 y t 6admperyzendigo.bg t y 43 Gb parallel 1 m y 4 y 4 4 46 4 y 12,4 12 48 4xt52 4 Day 3 Sec 1.2: Definitions 1. I can describe the undefined terms using words, diagrams, and notation. o Point o Line o Plane Term Definition Point A location. It has no size, and is shown by a dot Line Plane Image Notation p pointp p p any2points um A line has one dimension. It extends infinitely. hM A plane has two dimensions. It is represented by a shape that looks like a floor or a wall, but extends without end. moi line f P determine a EI rip in P E planeMNP p planer M P and Collinear Coplanar 7 Two or more points that lie on the same line q op T Three or more points that lie on the same plane f Parallel Perpendicular Two lines that intersect each other at right angles h T g v Tis non coplanar in Am L non collinear to more than one of them M NiP are coplanar k p L are collinear unandpareparanee una Two lines in the same plane that never intersect line p Ilp unandpareperpendicular un 1 P 5 1. Name line PT in four different ways. in 2. Name the Plane two different ways. SF FT ET line plane VST plane R 3. Name three points that are collinear. S P T 4. Name four points that are coplanar. PTV S 5. Name two points that are not collinear. Q T 6. Name two points that are not Coplanar. QS Parallel Lines and Skew Lines Parallel Lines (Coplanar lines that don’t intersect) Skew Lines (Non-Coplanar lines that don’t intersect) B E D A D A C C It appears that these lines do intersect but it is a distortion in the 2-D drawing of the 3-D relationship - AB and CD will never intersect. Ex 7 Determine the relationship between the lines. a) AB and CD Skew O c) and Parallel AF EF Skew Parallel b) EF and DG Intersect Skew d) CG and AH Skew Parallel 00 Intersect O Parallel Intersect Intersect B A C D H G E F 6 Sec 1.2 Definitions 2. I can define the following defined geometric terms using words, diagrams, and notation: o Line Segment o Ray Sec 1.3: Algebra Applications in Geometry 3. I can apply the segment addition property to solve problems Day 4 We use lines to define the following terms: Term Definition Image Notation A part of a line, having two endpoints Line Segment Has definite Length A Endpoint The point where a line segment or ray ends endigot Ray A part of a line that has a fixed start point, but no endpoint Two figures that are the same shape and size AB endpoint engepoint B A A 5 or BA nor angnapoint B A AB B Two rays with a common endpoint that extend in opposite directions, forming a L o D straight line Opposite Rays Congruent C B RQ R Q DF a F E B a c 7 AI E CT D 1. Convert the mathematical symbols to words. a) XY c) XY _______________________ XY b) segment _______________________ of segment d) the length e) XYZ _______________________ XYZ angle XY LL XY _______________________ line XY x XY _______________________ XY ray f) mXYZ measure _______________________ ofangleXYZ 2. When do you use congruent? When do you use equal? Give an example. a 3. measure when you're talking specific length or6mm the it's AB about that andsize AI of same Why can segments shapebe congruent BUT rays and lines cannot be congruent? b c rays and lines don't haveendpoints at both ends no defined length 7 Segment Addition If B is between A and C, then AB+BC=AC If AB+BC=AC, then B is between A and C 1. 39126 uw 65units 2. 63 50 13 pE 3. Solve for x. Xt I 2 1 x 4. 2 3 2 1 X 2x 7 2 5. X W u I3uni 16 6 266 1462 s Challenge: The point R lies on the line segment PS, where PR = 5x +2 cm, RS = 10cm , and PS = 𝑥 2 + 4𝑥 + 6 cm. What will be the value of x, PR, and RS? 8 Day 5 Sec 1.1: Pre-req Targets • midpoint Sec 1.3: Algebra Applications in Geometry 4. I can write the equation of the perpendicular bisector of the segment given its endpoints. Term Definition Intersection Point The point where two or more figures meet. Image r The middle of a segment s og intersection L Midpoint Notation point I i B a M is themidpoint of AI E A geometric figure (line, point, line segment, or Segment Bisector ray) that cut a segment in two equal parts a I B i m isthe bisectorof AT EXAMPLES: 1. Midpoint M is the midpoint of AB . If AM = 4x + 2 and MD = 8x -10 , find AB. 4 A I 2 8 10 M 2 8 10 4xtio 4 it 0 4x 4 Tz 3 I 93 4Gt2t8C3 1O rtEffiI AB 2. Midpoint in the coordinate plane Determine the midpoint given the following two endpoints A (-3,4) B (17,-2) (𝑥 + 𝑥2 ) (𝑦1 + 𝑦2 ) ( 𝑥𝑚 , 𝑦𝑚 ) = ( 1 , ) 2 2 1 31172,42 21,22 7 9 3. Shoppe pmidpoint What is the equation of a Perpendicular bisector to ̅̅̅̅ 𝐴𝐵 where A (4,-1) and B (6,3)? 1st Find the slope of the segment perpendicular to ̅̅̅̅ 𝐴𝐵 . original 4 m 214 Im 2nd Find the midpoint of ̅̅̅̅ 𝐴𝐵 . 42 21 111 3rd Using the ⊥ 𝑚 and the midpoint, write the equation of the line. Write your final answer in slope-intercept form. 6,1 as 1 y y 1 y 4. Csi ya 2 426 5 42 71 yzxt3 ̅̅̅̅ where A (4,2) and B What is the equation of the perpendicular bisector to 𝑨𝑩 (8,6)? 10 Sec 1.1: Pre-req targets: • Distance Formula • Pythagorean Theorem Day 6 WARM-UP Notice & Wonder find the length of segment JK and LM units units Is What if we can’t count… Option 1: Pythagorean Theorem c b 3 Option 2: Distance Formula Given points A and B are points on the coordinate plane, then the distance between A and B is: 𝑫 = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐 𝒙𝟐 𝒚𝟐 a 4 𝑨𝟐 + 𝑩𝟐 = 𝑪𝟐 (𝟒)𝟐 + (𝟑)𝟐 = 𝑪𝟐 𝟏𝟔 + 𝟗 = 𝑪𝟐 𝟐𝟓 = 𝑪𝟐 √𝟐𝟓 = √𝑪𝟐 ±𝟓 = 𝑪 *choose positive* C=5 Distance = 5 units 𝒙𝟏 𝒚𝟏 𝑫 = √(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐 𝑫 = √(𝟓 − 𝟏)𝟐 + (𝟓 − 𝟐)𝟐 𝑫 = √(𝟒)𝟐 + (𝟑)𝟐 𝑫 = √𝟏𝟔 + 𝟗 𝑫 = √𝟐𝟓 𝑫 = 𝟓 units 11 1. find the length of the following segment: D E I'I C ii 39 xz x.ftcyz yj L4 d 35tc3 Tsofhl C 75TT D d 49T Go f25552 d A 552 5fzunit a2t.b c2 c's 12 72 49 CZ 1 550 0 S un 12 Day 7 Sec 1.2: Definitions 2. I can define the following defined geometric terms using words, diagrams, and notation: • Angle Sec 1.3 Algebra Applications in Geometry 5. I can apply the angle addition property to solve problems. 6. I can apply the definition of an angle bisector to solve problems. Term Definition An angle consists of two different rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex. Angle Image Notation L ABC LCBA N A B 3 3 By vertex a middeffier cane µ The point where the sides/edges of an object meet. Vertex Angle Bisector A ray that divides an angle into two angles that are congruent. vertex D.d a F A LC Ll BCA then LDEGELFEG t 1. Name the angle in four different ways. LACB if ET is theangle bisectorof LDEF C 1 B 2. Find the measure of the following angles. 46 O 1340 0 461 134 180 13 Types of Angles 𝒎∠𝟏 < 𝟗𝟎𝒐 𝒎∠𝟐 = 𝟗𝟎𝒐 𝟗𝟎𝒐 < 𝒎∠𝟑 < 𝟏𝟖𝟎𝒐 𝒎∠𝟒 = 𝟏𝟖𝟎𝒐 Angle Addition If P is in the interior of ∠RST, then the measure of ∠RST is equal to the sum of the measures of ∠RSP and ∠PST 𝒎∠𝑹𝑺𝑻 = 𝒎∠𝑹𝑺𝑷 + 𝒎∠𝑷𝑺𝑻 1. Find the 𝑚∠𝐺𝐷𝐹. 2. Given 𝑚∠𝐴𝐵𝐶 = 118°, find x, and the 𝑚∠𝐴𝐵𝐷. I2886 7 onLGDE 420 mLABD 2Gs t35 10 6s 14 Angle Bisector An Angle Bisector is a ray that divides an angle into two angles that are congruent. ⃗⃗⃗⃗⃗⃗⃗ bisects ∠XYZ. In the Diagram, 𝒀𝑾 ∠XYW ≌ ∠ZYW and m∠XYW = m∠ZYW ∠𝑅𝑆𝑇 is a right angle. If ⃗⃗⃗⃗⃗ 𝑆𝑈 bisects ∠𝑅𝑆𝑇, find 𝑚∠𝑅𝑆𝑈. 3. 90 Z R A N ago If ⃗⃗⃗⃗⃗⃗ 𝐵𝐷 bisects ∠𝐴𝐵𝐶 and 𝑚∠𝐴𝐵𝐶 = 116° and 𝑚∠𝐷𝐵𝐶 = 2𝑥 − 4°, find x. 4 58 g D A A 441160 4 of s.gr 900 mLRSU 4. 2 132 4 58 t 4 Ex 2 2 4 4 C 2 4 116 8 116 4x x 124 15