TIInterActive File(Il moto rettilineo uniforme e la retta nel piano cartesiano (si prevedono circa 8 ore di lavoro in classe) Prerequisiti: funzioni lineari, successioni lineari; concetto di funzione; differenze finite; polinomi e operazioni con essi. Premessa Nei primi due anni di corso abbiamo gi affrontato il problema dei moti e della loro rappresentazione (mediante tabelle numeriche, mediante grafici e formule). Abbiamo anche anticipato alcuni concetti come quello di derivata, che dovrebbero averti fatto capire, per esempio, che la velocit la derivata prima, rispetto al tempo, della funzione s=s(t) che descrive come varia la posizione s di un corpo rispetto al tempo t. All'inizio di questo nuovo anno di corso ci proponiamo di riprendere in considerazione cose che gi dovresti conoscere, per sistematizzarle e consolidarle. Questo lavoro ha un duplice obiettivo: a) da una parte introdurre l'uso di tecniche e strumenti analitici per la descrizione di situazioni cinematiche e geometriche e per la risoluzione di alcuni semplici problemi a esse associati; b) dall'altra preparare il terreno alla descrizione di moti che non hai ancora descritto formalmente e che caratterizzano il comportamento di uno dei modelli pi importanti nella fisica: l'oscillatore armonico. Nel lavoro teso a conseguire il primo dei due obiettivi appena menzionati avrai la possibilit di ripercorrere alcune pagine particolarmente significative nella storia del pensiero matematico: quelle che portarono alla nascita della geometria analitica dovuta, principalmente, a due grandi matematici dl XVII secolo, Ren Descartes e Pierre De Femat (per saperne di pi clicca sull'hotword il pensiero matematico del XVII secolo ). Scheda 1. Moti rettilinei (Si prevedono circa 2 ore di lavoro in classe) Quando si dice che un corpo si muove di moto rettilineo, vuol dire che la sua traiettoria, ossia l'insieme dei punti che effettivamente percorre, una retta. Sembra di poter dire che il moto rettilineo sia il moto pi semplice da riprodurre; sar anche il moto pi facile da descrivere, ma trovare un sistema utile per guidare l'asta del pistone di una macchina a vapore in un moto rettilineo ha costituito un grosso problema per gli ingegneri che lavorarono negli anni a cavallo tra il XVIII e il XIX secolo (per saperne di pi clicca sull'hotword meccanismo di Watt ). Nel 1784 James Watt, l'inventore della macchina a vapore, ottenne una soluzione pratica e approssimata del problema mediante un meccanismo costituito da tre aste incernierate, due delle quali, DA e BC, di eguale lunghezza e una AB molto pi corta. Se D e C sono punti fissati (ad altezze opportune diverse fra loro) allora, facendo ruotare l'asta AD di un piccolo angolo, il punto medio P descrive per un tratto considerevole un segmento rettilineo verticale. Puoi vedere un'animazione del meccanismo di Watt aprendo il seguente file di Cabri II plus o, se non hai Cabri II plus, puoi cliccare sul corrispondente file in cabrijava: wattcabriIIplus wattcabrijava Oppure puoi collegarti a uno dei seguenti siti, che trattano il problema: Simulazione meccanismo di Watt sito del prof. Verdiani Meccanismo di Watt sul sito del "Giardino di Archimede" Una soluzione esatta del problema di generare con un meccanismo articolato un moto rettilineo fu trovata nl 1864 da Paucellier che costru un meccanismo piuttosto complesso, formato da sette aste incernierate; basandosi su una propriet che caratterizza una trasformazione geometrica detta inversione circolare, il meccanismo di Paucellier trasforma un moto circolare in un moto rettilineo. Se vuoi saperne di pi sul meccanismo di Paucellier ti suggeriamo di collegarti al sito del museo delle macchine matematiche di Modena . Diciamo quindi rettilineo un moto la cui traiettoria una retta. Questa una caratterizzazione che fa riferimento a propriet geometriche, legate al luogo di punti descritto dal corpo in moto. Ci sono, per, considerazioni di carattere cinematico che rivestono fondamentale importanza e che riguardano le funzioni che esprimono le leggi orarie (d'ora oin avanti, pi brevemente, "funzioni orarie"), ossia la variazione della posizione, della velocit e dell'accelerazione nel tempo. Un corpo che si muove su una guida rettilinea (per esempio un treno su un percorso senza curve) percorre una retta e, quindi, si muove di moto rettilineo. La sua velocit, per, pu cambiare, cos come la sua accelerazione. Invece un corpo sul quale non agiscono forze, se percorre una traiettoria rettilinea, deve muoversi a velocit costante (provate a spiegare perch richiamando le conoscenze che avete di fisica). La nostra spiegazione La nostra spiegazione dopo il confronto con altri gruppi Vogliamo allora caratterizzare, non solo geometricamente, ma anche cinematicamente un corpo che si muova a velocit costante nel tempo: in tali casi si dice che il corpo si muove di moto rettilineo uniforme, intendendo che il suo vettore velocit costante (ossia che la traiettoria una retta e il valore della velocit non varia nel tempo). D'ora innanzi parleremo di punto P per indicare il moto di un corpo P. Se il valore della velocit del punto P non varia nel tempo, abbiamo che v(t) = costante. Quindi, nel piano (t,v) il grafico della funzione oraria della velocit di P una retta orizzontale. Domanda 1: quale la funzione oraria dell'accelerazione del punto P? La nostra risposta (motivata) Domanda 2: quale la finzione oraria della posizione di P (fate l'ipotesi che il moto possa essere decritto da un'unica coordinata, ossia avvenga in una dimensione)? (Suggerimento: si tratta di trovare una funzione la cui derivata sia una costante ....) La nostra risposta (motivata) Fino a ora vi siete occupati del moto rettilneo uniforme in una dimensione e avreste dovuto concludere che le equazioni delle funzioni orarie sono le seguenti: x(t) = k * t + x0 , con k che indica la velocit (costante) v(t) = k a(t) = 0 Notate che x, v e a sono funzioni lineari del tempo. Che cosa accade se il moto rettilineo uniforme avviene nel piano? In tal caso il punto P viene individuato da due coordinate (x; y), ciascuna delle quali varia linearmente nel tempo: x(t) = k * t + x0 vx(t) = k ax(t) = 0 e y(t) = h * t + y0 vy(t) = h ay(t) = 0 dove k e h sono le, rispettivamente, le velocit orizzonale e verticale, ossia le proiezioni, lungo l'asse x e lungo l'asse y, del vettore velocit. Pu essere interessante cercare di determinare, dalle due funzioni orarie che esprimono la variazione di x e y in funzione del tempo, l'equazione della traiettoria, ossia della retta del piano che il punto P(x; y) percorre. Ci pu essere fatto risolvendo il sistema formato dalle due equazioni parametriche (il tempo t il parametro rispetto al quale viene descritta la variazione della x e della y del punto): x(t) = k * t + x0 e y(t) = h * t + y0 Trovare la traiettoria vuol dire trovare la dipendenza esplicita di y da x, eliminando quindi il tempo dalla prima equazione:  e sostituendolo nella seconda:  Ottenendo cos una funzione lineare di pendenza h/k e di intercetta (-hx0+ky0)/k. La traiettoria di un corpo P che si muove, senza vincoli, con velocit di valore costante, una retta, la cui pendenza data dal rapporto dei valori delle due velocit h (verticale, lungo y) e k (orizzontale, lungo x). Nella seguente animazione con CabriIIplus potete osservare la traiettoria di un punto P le cui coordinate x e y si muovono, al variare del tempo, di moto rettilineo uniforme. La funzione oraria dell'ascissa x = vx*t e quella dell'ordinata y = vy *t con vx e vy costanti. Se siete abituati a utilizzare cabriIIplus, aprite il file, fate qualche esplorazione modificando vx e vy con l'apposito cursore e poi provate a dire come stato costruito il foglio di lavoro. motorettilineouniformeparametricoplus Se non avete CabriIIplus, limitatevi a un'esplorazione del file di cabrijava motorettilineouniformeparametricojava . Il risultato delle nostre esplorazioni (Se siamo soliti utilizzare Cabri II plus) Come stato costruito il file di Cabri II plus sul moto rettilineo uniforme? Domanda 3: che cosa vi aspettate che accada per un punto P che si muova nello spazio di moto rettilineo uniforme? Provate e a scrivere le funzioni orarie delle sue coordinate e a trovare l'equazione della sua traiettoria. Cercate quindi un'interpretazione geometrica per le equazioni che avete ottenuto e alcuni dei loro parametri. La nostra risposta (motivata e con eventuale esplicitazione di punti non chiari) Abbiamo visto due modi differenti di rappresentare il moto di un punto che, per esempio, si muove nel piano: a) si danno le funzioni orarie della sua ascissa e della sua ordinata. In tal caso si dice anche che si danno le equazioni parametriche delle sue coordinate (ossia si precisa come ascissa e ordinata variano al variare del parametro t); b) si d l'equazione della sua traiettoria, ossia si precisa la relazione esistente tra le ascisse e le ordinate del luogo dei punti del piano descritto da P. Il primo punto di vista (quello di dare le equazioni parametriche) pi legato agli aspetti cinematici; il secondo punto di vista (quello di dare l'equazione cartesiana della traiettoria) pi geometrico. Naturalmente se il moto unidimensionale i due approcci si confondono: da questo punto di vista hanno quindi interesse moti che avvengano almeno nel piano. Notate che, nel caso del moto rettilineo uniforme, sia le equazioni parametriche, sia quella cartesiana, esprimono funzioni lineari e quindi i loro grafici sono rette. Scheda 2. La retta nel piano cartesiano (Si prevedono circa 6 ore di lavoro in classe) Proseguiamo il nostro discorso, iniziato con lo studio del moto rettilineo uniforme, con lo studio della retta dal punto di vista della geometria analitica del piano. Richiamiamo tre conoscenze fondamentali per la geometria analitica piana, ossia quella parte della matematica che si propone di utilizzare il potente strumento numerico e algebrico per descrivere oggetti geometrici del piano, per risolvere problemi di geometria piana e per dimostrare propriet geometriche delle figure del piano: blc <P=P?/PnPo~PPPPP&PP@PW@P& 8 9 |@P   @P   O@P$ 6Z [ @P] 7  @P{*@P1PG@PN8Pj@P P>@P7PU@P[ PdPZ@P\Pz@P5@P:E@P@P@P@P@P@P@Ps@Pt@Pv@Pw @P@P@P@P@P@P @P@PO@P@P@P@P@P@P!@P@P @P@P@P@Pn@P_@P`@Pd@PeY@PP%PNP4% Y ZP]P P P D@PY!PP!%@P%'P%@P%$@Pp"Arial"Arial"Arial"Arial"Arial"Arial<==>>?nocd$%     " # [ \   /001FGGHHIIJJKKLLMMN,--.wx5667TUUVVWWXXYYZZ[YZZ[[\z{{||}}~~$%abjkstabbctu~MN  23[\\]                W!X!X!Y!!!!!!!!!!!!!!!!!!!!! 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ffff0000 ffff ffff0000 1 1 0 0 0 130000 804e 0 0 0 10000 1 0 0 804c29 0 0 0 ff000000 1ff 0 0 0 0 13000000 300 5c005500 5600 0 0 1804c00 43000000 ff000001 ffffff ff000000 1ff 100 0 0 0 13000000 1900 1900 1900 1900 0 50300 17f00 ffffff00 ffffffff 1ffffff 1000000 0 0 0 1000000 0 3000000 5000000 53454d43 254 20de0280 0 0'33 10001 0 0 0 0 3601 0 10000 0 0 0 0 0d@@y= (h*(x - x0))/(k) + y0Times New Roman y= (h*x - h*x0 + k*y0)/(k)Times New Roman   NANIYi9http://matematica.uni-bocconi.it/galeazzi/capitolo11.htmNhttp://www2.math.unifi.it/~archimede/archimede/curve/visita/virtuale.php?id=1Ihttp://www.matematica.it\paola\Terzoanno\Geometriaanalitica\wattplus.fig1http://www.matematica.it\paola\Cabriweb\watt.htm8http://www.matematicamente.it/approfondimenti/macchine/Ehttp://www2.math.unifi.it/~archimede/archimede/curve/geomeccan3.html$http://www.museo.unimo.it/theatrum/Vhttp://www.matematica.it\paola\Terzoanno\Geometriaanalitica\rettilienoparametrico.fig Bhttp://www.matematica.it\paola\Cabriweb\rettilineoparametrico.htmArialArialArialArialArialArial<a) i punti sono coppie ordinate di numeri reali. In un fissato sistema di riferimento, una coppia ordinata (x; y) di numeri reali individua uno e un solo punto P del piano e, viceversa, ogni punto del piano individua, in un fissato sistema di riferimento, una e una sola coppia ordinata di numeri reali; b) in un fissato sistema di riferimento ogni retta che non sia verticale descritta da un'equazione lineare in x e y, ossia da un'equazione del tipo: y = p*x + q. Ricordiamo che il coefficiente p indica la pendenza della funzione lineare che ha per grafico la retta e il coefficiente q indica l'intercetta, ossia l'ordinata del punto di ascissa nulla. Le coordinate dei punti che appartengono alla retta possono essere interpretate come soluzioni dell'equazione lineare (nelle due incognite x e y) y = p*x+q . Ci si esprime formalmente con la notazione {(x;y) : y = p*x+q}. La retta quindi un sottoinsieme del piano. Le rette verticali sono invece descritte da equazioni del tipo x = k, con k costante (sono infatti i luoghi geometrici dei punti che hanno stessa ascissa). Naturalmente, come ben sai, le equazioni del tipo x = k non sono funzioni lineari del tipo y = f(x); c) in ogni sistema di riferimento cartesiano, la distanza fra due punti si calcola utilizzando il teorema di Pitagora. Domanda 4: nel precedente punto c) si detto che la distanza fra due punti si calcola utilizando il teorema di Pitagora. Provate a calcolare la distanza tra i due punti A(2; 1) e B (3; 5) (suggerimento: considerate il triangolo rettangolo AHB dove H il punto di intersezione tra la perpendicolare condotta per B all'asse delle ascisse e la perpendicolare condotta per A all'asse delle ordinate) La nostra risposta Domanda 6: provate a generalizzare il procedimento prima utilizzato. Considerate ora due punti A (xA; yA) e B (xB; yB) . Come possibile calcolare la distanza AB, ossia la lunghezza del segmento AB in funzione delle coordinate di A e B? La nostra risposta Domanda 7: se, per, i punti hanno stessa ascissa o stessa ordinata, ancora necessario utilizzare la formula generale ricavata utiizzando il teorema di Pitagora? Provate con qualche caso particolare, per esempio con i punti A(2; 1) e B(2; 5) e con i punti C(3; 8) e D(0.5; 8). La nostra risposta Eventuali punti non chiari relativi al calcolo della distanza fra due punti di cui sono note le coordinate Con le seguenti quattro conoscenze fondamentali della geometria analitica del piano: a) un punto una coppia ordinata di numeri reali; b) una retta non verticale rappresentata da un'equazione del tipo y = p*x + q e una retta verticale rappresentata da un'equazione del tipo x = k; c) dati due punti A (xA; yA) e B (xB; yB) , la distanza AB (ossia la lunghezza del segmento AB) si calcola con la formula:  oppure con le formule |yA - yB| se A e B hanno stessa ascissa e |xB - xA| se A e B hanno stessa ordinata d) la pendenza di un segmento AB la stessa pendenza della retta AB e rette parallele hanno stessa pendenza. siamo pronti per iniziare a descrivere analiticamente le figure del piano. Inizieremo da alcune rette interessanti e dalla relazioni di parallelismo e perpendicolarit tra rette. Asse di un segmento Incominciamo con il domandarci come sia possibile trovare l'equazione dell'asse di un segmento AB di cui si conoscono le coordinate (a; b) di A e (c; d) di B. Se pensiamo all'asse del segmento AB come alla retta passante per il punto medio di AB e perpendicolare ad AB, ci complichiamo la vita, perch dovremmo innanzitutto trovare una formula che dia le coordinate del punto medio di un segmento e una formula che esprima la relazione di perpendicolarit tra due rette. Conviene utilizzare un'altra propriet dell'asse di un segmento: l'asse di un segmento l'insieme di tutti e soli i punti del piano che sono equidistanti dagli estremi del segmento. Si tratta di una propriet che chiama in causa la distanza fra punti, che sappiamo come calcolare. Sar quindi sufficiente indicare con P(x; y) le coordinate dei punti appartenenti all'asse e imporre che la distanza PA sia uguale alla distanza PB. In formule, PA = PB. Si tratta ora di applicare la formula della distanza fra due punti. Risolviamo insieme il problema facendosi aiutare da TI-InterActive!: PA =  PB=  Quindi, PA=PB implica:  Si tratta ora di semplificare l'espressione; poich si tratta dell'uguaglianza tra due termini entrambi positivi e poich i radibcando sono sicuramente non negativi (perch?), possiamo elevare a quadrato entrambi i membri e ottenere un'equazione equivalente senza le radici:  E, semplificando i termini simili, si ottiene 2(c - a)*x + 2(d - b) * y + a^2 + b^2 - c^2 - d^2 = 0 ossia l'equazione di una retta, come del resto ci si attendeva. In particolare, se d - b = 0, ossia se d = b, allora la retta ha equazione  altrimenti otteniamo  Ossia l'equazione di una funzione lineare di pendenza  Domanda 8 (attivit individuale) Determina l'equazione dell'asse di ciascuno dei seguenti segmenti: A (6; 0.4) B(-3; 2.1) C(1/3; -1/2) D(2; 4) E(3; 4) F(1.5; 2) La mia risposta Punto medio di un segmento e relazione di perpendicolarit fra rette L'esercizio di determinazione dell'equazione dell'asse di un segmento ci ha portato a risolvere, senza volerlo, altri due problemi: a) la determinazione delle coordinate del punto medio di un segmento di estremi A(a; b) e B(c; d) b) la determinazione della relazione di perpendicolarit fra due rette. Se, infatti, pensiamo all'asse di un segmento AB come alla retta perpendicolare alla retta AB e passante per il punto medio del segmento AB, dato che abbiamo risolto in generale il problema della determinazione dell'equazione dell'asse di AB, possiamo risolvere in generale i due problemi ora posti. Vediamo come, iniziando dal secondo, ossia dalla determinazione della relazione di perpendicolarit fra due rette (supponiamo che la retta AB non sia n verticale, n orizzontale, in quanto in tali casi o la pendenza di AB, o quella del suo asse non sarebbero definite). La retta AB ha come pendenza la pendenza del segmento AB e, quindi (d-b)/(c-a) oppure, equivalentemente, (b-d)/(a-c). Se guardiamo l'equazione dell'asse di AB, notiamo che la pendenza di tale asse (a-c)/(d-b), ossia esattamente il reciproco dell'opposto della pendenza di AB. Possiamo quindi dire che la relazione che lega la pendenza p' dell'asse di AB a quella p di AB la seguente p*p' = - 1 D'altra parte ogni retta perpendicolare ad AB ha la stessa pendenza dell'asse di AB (infatti tute le rette perpendicolari ad AB sono parallele all'asse di AB e, quindi, hanno stessa pendenza) e quindi possiamo concludere che la relazione p*p' = - 1 lega fra loro le pendenza p e p' di due rette fra loro perpendicolari. L'intersezione dell'asse del segmento AB con la retta passante per A e per B, d'altra parte, consentir di ottenere le coordinate del punto medio M del segmento AB, dato che l'asse di AB passa per M. Possiamo quindi farci aiutare da TI-InterActive! per determinare l'intersezione tra l'asse del segmento AB e la retta passante per A e B. Iniziamo con il determinare l'equazione della retta passante per AB. Supponiamo che tale retta non sia verticale, ossia che le ascisse a e c dei punti A e B siano fra loro diverse. In tal caso, la retta AB avr equazione del tipo y = p * x+ q. La pendenza p data dal rapporto (b-d)/(a-c). Per determinare l'intercetta q sufficiente imporre che le coordinate di A siano soluzioni dell'equazione:  Quindi:  Sa cui, risolvndo rispetto a q si ottiene:  Per determinare l'ascissa del punto medio del segmento AB quindi sufficiente risolvere la seguente equazione:  Come potete vedere si ottine come soluzione x = (a+c)/2, ossia la media aritmetica delle ascisse dei punti A e B. La seconda condizione afferma che la soluzione x = (a+c)/2 vale se il segmento AB non verticale n orizzontale, ossia se la pendenza del segmento AB e quella del suo asse esistono ( e  ). Per determinare l'ordinata del punto medio sufficiete sostituire il valore dell'ascissa, ossia (a+c)/2 al posto della x nell'equazione della retta AB (o in quella del suo asse):  Come potete notare, l'ordinata del punto medio M del segmento AB la media aritmetica delle ordinate. Quindi il punto medio M di un segmento AB, ove A(a;b) e B(c; d), M((a+c)/2; (b+d)/2). Eventuali parti non comprese o poco chiare: Domanda 9. Il fatto che le coordinate del punto medio di un segmento AB siano la media aritmetica delle ascisse e delle ordinate degli estremi A e B non dovrebbe stupirvi moto: diciamo che un risultato che avreste potuto aspettarvi. Provate a ottenere questo risultato per altra via (suggerimento: osservate con attenzione la seguente figura... non notate una configurazione che richiama il teorema di Talete?)  La nostra risposta Questo secondo approccio con cui avete determinato le coordinate del punto medio di un segmento non solo pi economico e veloce, ma anche suscettibile di generalizzazione. Come potreste, infatti, generalizzare il problema per dividere un segmento non in due parti, ma in tre parti uguali? E in cinque? E in n parti uguali? La nostra risposta La distanza di un punto da una retta In alcuni problemi pu esere utile avere una formula che consenta di determinare la distanza di un punto da una retta. Pensate, per esempio, che l'area di un triangolo si determina dividendo per 2 il prodotto fra una base e l'altezza a essa relativa e che tale altezza pu essere pensata come la distanza dalla base considerata dal vertice a essa opposto (poich tutti i poligoni possono essere scomposti in triangoli, una procedura che consente di determinare l'area di un triagolo consente di risolvere il problema di determinare l'area di un qualunque poligono). Dati del problema: a) punto A(a; b) b) equazione della retta r: y = p*x+q (supponiamo che la retta sia non verticale; il problema di determinare la distanza di un punto da ua retta verticale semplice) Richiesta: formula che consente di calcolare la distanza di A da r. Vi chiediamo di completare, aiutandovi anche con TI-InterActive!, la seguent procedura che consente di risolvere il problema. Passi della procedura: 1. determinazione della retta t passante per A e perpendicolare a r 2. determinare il punto H di intersezione tra t e r 3. calcolare la distanza AH che la distanza di A da r 1. L'equazione della retta t passante per A e perpendicolare a r : la pendenza di t -1/p. Quindi la retta t ha equazione del tipo y = -x/p + q' L'intercetta q si pu determinare imponendo che A sia soluzione dell'equazione y=-x/p + q' Il calcolo da noi effettuato per trovare l'equazione della retta t: Dovreste aver trovato un'equazione di t equivalente alla seguente: y = -x/p + (bp+a)/p 2. Calcolate ora le coordinate del punto H di intersezione fra t e r La nostra risposta Risolvendo il sistema tra le equazioni delle rette t e r dovreste trovare per H le seguenti coordinate: H( ; ) 3. Calcoliamo ora la distanza tra il punto A(a; b) e il punto H, ossia la distanza AH fra il punto A e la retta r:  Possiamo quindi concludere che la distanza di un punto A(a; b) da una retta di equazione y = p*x+q data dalla formula:  Domanda 10. (Attivit individuale). Calcola la distanza di ciascuno dei seguenti punti dalla retta di equazione a fianco indicata 1. A(3; 2) y = 3x -2 2. B(-4; 1/2) y = 0.5x + 1/2 3. C(1/2; -1/3) y = -x/3 - 2/5 La mia risposta Domanda 11. Calcola l'area dei poligoni definiti dai vertici precisati in ciascuno dei seguenti esercizi: 1. A(0;0) B(-1; 3) C(1/2; -4) 2. A(3;4) B(1; 1) C(-2;3) D(-1;8) La mia risposta Le equazioni delle bisettrici degli angoli formati da due rette r e t La formula della distanza di un punto da una retta consente di determinare le equazioni delle bisettrici degli angoli formati da due rette r e t incidenti. Notiamo subito che se le rette sono gli assi cartesiani, il problema facilmente risolvibile. Infatti le bisettrici degli assi cartesiani sono il luogo di tutti e soli i punti che hanno le ordinate uguali, in valore assoluto, alle ascisse. Quindi y=x la bisettrice del primo e terzo quadrante e y=-x quella del secondo e quarto quadrante.  Il problema altrettanto semplice se si vogliono determinare le equazioni delle bisettrici di rette parallele agli assi cartesiani. Domanda 12 Determina l'equazione delle bisettrici delle rette di equazione x= 2 e y = -4 La mia risposta Domanda 13 Determina l'equazione delle bisettrici delle rette di equazione x= h e y = -k La mia risposta Occupiamoci ora del problema un po' pi complesso e cio quello di determinare le equazioni delle bisettrici degli angoli formati da due rette r e t incidenti aventi equazione del tipo: r: y=px+q t: y=p'x+q' Per risolvere questo problema conviene pensare le bisettrici degli angoli formati da due rette incidenti come l'insime di tutti e soli i punti equidistanti dai lati dell'angolo considerato. cos possibile utilizzare la formula della disatnza di un punto da una retta. Consideriamo quindi un punto generico della bisettrice dell'angolo considerato, chiamandolo P(x; y) e applichiamo la formula per trovare la distanza PH di P dalla retta r  e la distanza PK di P dalla retta t:  A questo punto possibile uguagliare le due distanze, ottenendo le equazioni delle due bisettrici (fra loro perpendicolari) degli angoli formati da due rette incidenti:  Il seguente file di Cabri consente di fare qualche esplorazione sul problema ora affrontato (se non avete cabriII esplorate con il file di Cabrijava) bisettricicabriII bisettricicabrijava Domanda 14 1. Determina le equazioni delle bisettrici formate dalle due rette di equazione: y = x + 3 e y = x/2 + 4 2. Determina l'incentro (ossia il punto di incontro delle bisettrici) dei triangolo individuati dalle seguenti terne di punti: a) A(0;0) B(-1; 3) C(1/2; -4) b) A(3;4) B(1; 1) C(-2;3) 3. Determina il centro della circonferenza inscritta nel triangolo ABC ove A(2; 1) B(3; 4) C(1/2; 3) Abbiamo visto che la geometria analitica offre metodi molto potenti per la risoluzione dei problemi di geometria; la comprensione delle procedure e delle tecniche messe a disposizione dalla geometria analitica non difficile, ma per far proprie certe tecniche necessario molto esercizio, svolto anche insieme a studenti pi esperti e al proprio insegnante. Per questo motivo vi invitiamo a seguire con attenzione le lezioni in classe in cui il vostro insegnante risolver problemi di complessit maggiore di quelli che abbiamo proposto e utiizzer la geometria analitica per verificare o dimostrare propriet geometriche. Certi automatismi, importanti dopo che si capito, si acquisiscono anche per imitazione degli esperti. Prima di eseguire il solito test si autovalutazione, gli esercizi di consolidamento e le prove di verifica, ti chiediamo di fare tu, insieme a qualche compagno di gruppo, il documento di sintesi teorica di questa lezione. Confrontalo poi con quello proposto da altri compagni di gruppo e, infine, proponete la vostra sintesi all'insegnante. La nostra sintesi Test di autovalutazione Consolidamento Esempi di verifiche F@P P!P@PP @P PZP(P)P,P-P5P6P9P:yP@PP@P PPPP kPm Pu Y@P P @P P P P P P P P P mP- P. P2 P3 &PY PZ P^ P_ P @P P PV @PK@PL@PM@PN@PO@PP@PR@PS@PX@PY@PZ@P[@P\@P]@Pa@Po@Pq@Pv@Px@PV@P@P@P@P@P@P @P@P@P@Pw@Pf PphP<PP<P$P PP!IPjP)P+PEPGP[P\P]P^Pb<PcPVPWPXPYP]<P^P|P}PPPPPP[PPPPPPPP?P>P?xPPPgPhtPPPPP ,P!P$! 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1 0 80100003 1e 8f 71 0 ffff 10000 0 10200000 2d80 8f00 7100 0 ffff00 1000000 0 2d000000 1e8010 8f0000 710000 0 ffff0000 0 1 0 20 0 0 0 3c800d00 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 3000000 100 100 0 6400 0 0 13 802f0000 1 8f ffffffff 0 1ffff 10000 0 0 0 0 80310013 37 8f ffffffff 0 10001 10000 0 2f290000 180 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 1001300 4000000 5011301 1802f01 96000000 ff000000 ffffff ff000000 1ff 100 0 0 0 13000000 803100 0 0 0 1000000 100 0 0 802f2900 0 0 0 0 1ffff 0 0 0 0 0 30013 e90000 ea0116 0 0 80140000 80 8f 71 0 1ffff 10000 0 10000 80800b 8f0000 710000 0 ffff0000 0 1 0 80100003 1e 8f 71 0 ffff 10000 0 10200000 4480 8f00 7100 0 ffff00 1000000 0 3d000000 1e8010 8f0000 710000 0 ffff0000 0 1 0 20 0 0 0 3c803b00 8f000000 71000000 0 ff000000 ff 100 0 b000100 3c80 8f00 7100 0 ffff00 1000000 0 1000000 3c801000 8f000000 71000000 0 ff000000 1ff 100 0 3000 0 0 13 802f0000 1 165 ffffffff 0 1ffff 10000 0 0 0 0 650013 650000 650000 650000 0 a1c0000 1a10000 ffff0000 ffffffff ffffffff 1 1 0 0 0 1 0 3 454d4305 25453 de028000 20 0'33 10001 0 0 0 0 3601 0 10000 0 0 0 0 0d@@}solve( (2*(a - c)*x - (a)^(2) - (b)^(2) + (c)^(2) + (d)^(2))/(2*(d - b)) - (b - d)/(a - c) *x + (b*c - d*a)/(a - c) =0, x)Times New Roman `x= (a + c)/(2) or ((a)^(2) - 2*a*c + (b)^(2) - 2*b*d + (c)^(2) + (d)^(2))/((a - c)*(b - d)) =0Times New Roman   00 801f00 0 0 0 1000000 100 0 0 80092800 36 8f 71 0 ffff 10000 0 10000 36800b 8f0000 710000 0 ffff0000 0 1 0 800d0002 35 8f 71 0 ffff 10000 0 10000 35800b 8f0000 710000 0 ffff0000 0 1 0 80100001 35 8f 71 0 10003 10000 0 630000 0 0 1300 2d000000 180 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 2f001300 80 0 0 0 1000100 0 0 29000000 802d 0 0 0 ffff0000 1 0 0 0 0 130000 0 134012d 801c012e 2e 69 53 0 ffff 10000 0 10000 800b 0 0 0 ffff0000 0 1 0 801f0004 0 0 0 0 10001 0 0 9280000 2e80 6900 5300 0 ffff00 1000000 0 1000000 2e800b00 69000000 53000000 0 ff000000 ff 100 0 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1ffff 10000 0 320000 0 0 1c00 26000000 180 8f00 ffffff00 ff 1ffff00 1000000 0 0 0 0 28001c00 80 0 0 0 1000100 0 0 29000000 8026 0 0 0 ffff0000 1 0 0 0 0 1c0000 3 37002f 30 0 0 18026 8f0000 ffff0000 ffff ffff0000 1 1 0 0 0 1c0000 d d d d 0 221 178 ffffffff ffffffff 1ffff 10000 0 0 0 10000 0 30000 43050000 5453454d 80000002 20de02 0'33 10001 0 0 0 0 3601 0 10000 0 0 0 0 0d@@7( (b - d)/(a - c) *(a + c))/(2) - (b*c - d*a)/(a - c)Times New Roman  (b + d)/(2)Times New Roman    bdnKR$T pendenzadi un segmento AB la stessa pendenza della retta AB e rette parallele hanno stessa pendenza. siamo pronti per iniziare a descrivere analiticamente le figure del piano. Inizieremo da alcune rette interessanti e dalla relaNANIte. Conviene utilizzare un'altra propriet dell'asse di un segmento: l'asse di un segmento il luogo geometrico di tutti e soli i punti del piano che sono equidistanti dagli estremi del segmento. Si tratta di una propriet che chiama in causa la distanza fra punti, che sappiamo come calcolare. Sar quindi sufficiente indicare con P(x; y) le coordinate dei punti appartenenti all'asse e imporre che la distanza PA sia uguale alla distanza PB. In formule, PA = PB. Si tratta ora di auf#@dd;7 CTxobjItemࡱ>  Root Entry{NW` T^@ Contents OlePres0004  !"#$%&'()*+,-./0123,,?$@$?$@$?@|4!/l?-DT!@es-8R?-DT!@es-8R? 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