Objective: Use the Pythagorean Theorem and its converse to solve right triangle problems. CA Geometry Standard: 12, 14, 15

Transcription

1 Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 1 of 2 Ojective: Use the Pythgoren Theorem nd its converse to solve right tringle prolems. CA Geometry Stndrd: 12, 14, 15 Historicl Bckground Pythgors ws Greek mthemticin orn round 569 BC nd died round 475 BC. Although he is very fmous, we know little out his chievements. He ws the leder of society, hlf religious nd hlf scientific, tht followed code of secrecy. His followers were known s mthemtikoi. One thing tht we do know is tht they discovered wht is known s the Pythgoren Theorem. Key Concept: The Pythgoren Theorem B In right tringle, where the legs re length nd, nd the hypotenuse is length c, the following eqution is true: C c A A proof of the Pythgoren Theorem: y c Emples: Use the Pythgoren Theorem to solve for ech vrile y m Key Concept: The Converse of the Pythgoren Theorem In tringle where c is the longest side, if = c, then it is right tringle.

2 Geometry CP Lesson 8.2 Pythgoren Theorem nd its Converse Pge 2 of 2 Key Concept: A Pythgoren Triple is three whole numers tht stisfy the Pythgoren Theorem eqution. List some common Pythgoren Triples: Emple: Determine whether ech set of mesures cn e those of right tringle. Then stte if they form Pythgoren Triple , 40, 41 7, 28, 29,,3 5 5 Emple: Determine whether ΔHEY is right tringle if its vertices re: H(2, 7) E(3, 6) Y(-4, -1) Emple: An irplne lnds t n irport 60 miles est nd 25 miles north of where it took off. How fr prt re the two irports? Key Concept: Pythgoren Inequlities If > c, then ΔABC is If < c, then ΔABC is Emple: Acute, otuse, right or not tringle? 4, 9, 12 8, 15, 23

3 Geometry CP Lesson 8.3 Specil Right Tringles Pge 1 of 2 Ojective: Use the properties of specil right tringles to solve prolems. CA Geometry Stndrd: 20 There re two types of specil right tringles in mth Tringles --90 Tringles legs re congruent hypotenuse hypotenuse short leg long leg These specil tringles hve properties tht mke finding missing sides lot quicker! Memorize these properties nd they will sve you lots of time. You ll hve less hedches nd hve much hppier life in generl. Key Concept: The Properties of --90 Tringle In tringle, the hypotenuse is times s long s ech leg. Emples: Solve for ech vrile. h 10 w m y k 14

4 Geometry CP Lesson 8.3 Specil Right Tringles Pge 2 of 2 Key Concept: The Properties of --90 Tringle In tringle, the hypotenuse is s long s the shorter leg nd the longer leg is times s long s the shorter leg. Emples: Solve for ech vrile. 10 h y 12 h ft A 40 foot long escltor rises from the first floor to the second floor of shopping mll. The escltor mkes ngle with the horizontl. How high ove the first floor is the second floor? The perimeter of n equilterl tringle is 39 cm. Find the length of n ltitude of the tringle.

5 Geometry CP Lesson 8-4: Trigonometry Pge 1 of 3 Lesson ojective: Find trigonometric rtios using right tringles. CA Geometry Stndrd: 18, 19 Trigonometry ws developed for use y stronomers nd surveyors to clculte distnce or height. A rtio of the lengths of sides of right tringle is clled trigonometric rtio. The three most common trig rtios re: Definitions for trig rtios in right tringle. o sin X = o cos X = o tn X = These trig rtios ONLY pply to the cute ngles of right tringle. SOH CAH TOA In ΔABC, wht re the trig rtios for A? o sin A = cos A = tn A = In ΔABC, wht re the trig rtios for C? A c o sin C = cos C = tn C = B C Emple 1: Find the sin, cos, nd tn rtios for D nd F D sin D = cos D = tn D = 6 10 E 8 F sin F = cos F = tn F = Emple 2: Find the missing sides, then find the trig rtios for ech cute ngle: 4 sin = cos = tn = sin = cos = tn = Use scientific clcultor (must e in DEGREE mode), to find these vlues rounded to 4 deciml plces. Compre them to the vlues you determined ove. o sin = cos = tn = o sin = cos = tn =

6 Geometry CP Lesson 8-4: Trigonometry Pge 2 of 3 H Emple 3: 13 E Y Find rtios: sin E = cos E = tn E = Use clcultor: sin = cos = tn = 6 Emple 4: X 6 Find rtios: sin X = cos X = tn X = Use clcultor: sin = cos = tn = Solving Trig Equtions o Step 1: Identify the plyers (Hyp? Opp? Adj?) o Step 2: Identify the trig function tht pplies to the plyers (SOH? CAH? TOA?) o Step 3: Set up n eqution nd solve Emple 5: Emple 6: Emple 7: o Emple 8: 48 Emple 9:

7 Geometry CP Lesson 8-4: Trigonometry Pge 3 of 3 Inverse Trigonometric Functions o Inverse trig functions re used to find the mesure of n ngle. This cn only e done using your clcultor. sin X = cos X = 1 m X = sin m X = cos tn X = m X = tn Emples: Find the mesure of ech ngle 1 1 sin W = 5 8 m W = cos X = m X = tn Y = 1.5 m Y = sin Z = 2 2 m Z = Solving for ngles using trig equtions. Step 1: Identify the plyers (Hyp? Opp? Adj?) Step 2: Identify the trig function tht pplies to the plyers (SOH? CAH? TOA?) Step 3: Set up n eqution, use n inverse trig function to solve A 60-foot rmp rises from the first floor to the second floor of prking grge. The second floor is 15.5 feet ove the second floor. Wht ngle does the rmp mke with the first floor?

8 Lesson ojective: Solve prolems involving elevtion nd depression ngles. CA Geometry Stndrd: 19

9