1 \startsection[reference=seccio:teorema-de-tales, title={Teorema de Tales}]
3 \startsubject[title={Preguntes}]
5 \startexercici[reference=exercici:Tales-directe] Determineu, sense fer servir el regle, la longitud dels segments que falten mitjançant el teorema de Tales.
8 \item $\overbar{OA} = 4,2 \unit{centi meter}$, $\overbar{AB} = 3 \unit{centi meter}$, $\overbar{OA'} = 3,4 \unit{cnti meter}$, $\overbar{OC} = 11,9 \unit{centi meter}$.
10 \startplacefigure[location=force, reference=fig:exercici-Tales-1, title={Figura a}]
11 \bTABLE[frame=off,align=middle]
16 % pàgina 58 del manual de TikZ
17 \coordinate[label=below:$O$] (O) at (0,0);
19 \coordinate (A) at (20:4.2);
20 \coordinate (B) at (20:7.2);
21 \coordinate (C) at (20:11);
23 \coordinate (D) at (20:11.4);
24 \coordinate (E) at (20:-0.4);
26 \coordinate (A2) at (-10:3.4);
27 \coordinate (B2) at (-10:5.83);
28 \coordinate (C2) at (-10:8.90);
30 \coordinate (D2) at (-10:9.3);
31 \coordinate (E2) at (-10:-0.4);
34 \draw (A) node[anchor=south east] {$A$};
35 \draw (A2) node[anchor=north east] {$A'$};
36 \draw (B) node[anchor=south east] {$B$};
37 \draw (B2) node[anchor=north east] {$B'$};
38 \draw (C) node[anchor=south east] {$C$};
39 \draw (C2) node[anchor=north east] {$C'$};
42 % Extenc les línies de A-A', ...
43 % pàgina 60 del manual de TikZ
44 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
45 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
47 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
48 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
50 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
51 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
53 % Dibuix les línies secants
54 \draw[very thick, color=blue] (E) -- (D);
55 \draw[very thick, color=blue] (E2) -- (D2);
57 % Dibuix els segments paral·lels
58 \draw[thick, color=blue] (F) -- (F2);
59 \draw[thick, color=blue] (G) -- (G2);
60 \draw[thick, color=blue] (H) -- (H2);
63 \foreach \point in {O,A,B,C,A2,B2,C2}
64 \fill [black] (\point) circle (2pt);
72 \item $\overbar{OB} = 7,8 \unit{centi meter}$, $\overbar{OC} = 12,3 \unit{centi meter}$, $\overbar{OB'} = 8,3 \unit{centi meter}$, $\overbar{B'C'} = 4,7 \unit{centi meter}$, $\overbar{OA} = 4,7 \unit{centi meter}$. Determineu primer $\overbar{OC'}$, $\overbar{BC}$ i $\overbar{OA'}$.
74 \startplacefigure[location=force, reference=fig:exercici-Tales-2, title={Figura b}]
75 \bTABLE[frame=off,align=middle]
80 % pàgina 58 del manual de TikZ
81 \coordinate[label=below:$O$] (O) at (0,0);
83 \coordinate (A) at (10:4.7);
84 \coordinate (B) at (10:7.8);
85 \coordinate (C) at (10:12.3);
87 \coordinate (D) at (10:12.7);
88 \coordinate (E) at (10:-0.4);
90 \coordinate (A2) at (-10:5);
93 \coordinate (B2) at (-10:8.3);
94 \coordinate (C2) at (-10:13);
96 \coordinate (D2) at (-10:13.3);
97 \coordinate (E2) at (-10:-0.4);
100 \draw (A) node[anchor=south east] {$A$};
101 \draw (A2) node[anchor=north east] {$A'$};
102 \draw (B) node[anchor=south east] {$B$};
103 \draw (B2) node[anchor=north east] {$B'$};
104 \draw (C) node[anchor=south east] {$C$};
105 \draw (C2) node[anchor=north east] {$C'$};
108 % Extenc les línies de A-A', ...
109 % pàgina 60 del manual de TikZ
110 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
111 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
113 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
114 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
116 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
117 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
119 % Dibuix les línies secants
120 \draw[very thick, color=blue] (E) -- (D);
121 \draw[very thick, color=blue] (E2) -- (D2);
123 % Dibuix els segments paral·lels
124 \draw[thick, color=blue] (F) -- (F2);
125 \draw[thick, color=blue] (G) -- (G2);
126 \draw[thick, color=blue] (H) -- (H2);
129 \foreach \point in {O,A,B,C,A2,B2,C2}
130 \fill [black] (\point) circle (2pt);
139 \item $\overbar{OA'} = 3,5 \unit{centi meter}$, $\overbar{A'B'} = 4,3 \\unit{centi meter}$, $\overbar{AB} = 6,3 \unit{centi meter}$, $\overbar{B'C'} = 2,2 \unit{centi meter}$.
141 \startplacefigure[location=force, reference=fig:exercici-Tales-3, title={Figura c}]
142 \bTABLE[frame=off,align=middle]
147 % pàgina 58 del manual de TikZ
148 \coordinate[label=below:$O$] (O) at (0,0);
150 \coordinate (A) at (20:5);
151 \coordinate (B) at (20:11.14);
152 \coordinate (C) at (20:14.28);
154 \coordinate (D) at (20:14.5);
155 \coordinate (E) at (20:-0.4);
157 \coordinate (A2) at (-10:3.5);
158 \coordinate (B2) at (-10:7.8);
159 \coordinate (C2) at (-10:10);
161 \coordinate (D2) at (-10:10.4);
162 \coordinate (E2) at (-10:-0.4);
165 \draw (A) node[anchor=south east] {$A$};
166 \draw (A2) node[anchor=north east] {$A'$};
167 \draw (B) node[anchor=south east] {$B$};
168 \draw (B2) node[anchor=north east] {$B'$};
169 \draw (C) node[anchor=south east] {$C$};
170 \draw (C2) node[anchor=north east] {$C'$};
173 % Extenc les línies de A-A', ...
174 % pàgina 60 del manual de TikZ
175 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
176 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
178 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
179 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
181 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
182 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
184 % Dibuix les línies secants
185 \draw[very thick, color=blue] (E) -- (D);
186 \draw[very thick, color=blue] (E2) -- (D2);
188 % Dibuix els segments paral·lels
189 \draw[thick, color=blue] (F) -- (F2);
190 \draw[thick, color=blue] (G) -- (G2);
191 \draw[thick, color=blue] (H) -- (H2);
194 \foreach \point in {O,A,B,C,A2,B2,C2}
195 \fill [black] (\point) circle (2pt);
204 \item $\overbar{B'D'} = 7,1 \unit{centi meter}$, $\overbar{A'C'}= 8,1 \unit{centi meter}$, $\overbar{AB} = 3,9 \unit{centi meter}$, $\overbar{BD} = 5,8 \unit{centi meter}$, $\overbar{OA} = 5,1 \unit{centi meter}$.
206 \startplacefigure[location=force, reference=fig:exercici-Tales-4, title={Figura d}]
207 \bTABLE[frame=off,align=middle]
212 % pàgina 58 del manual de TikZ
213 \coordinate[label=below:$O$] (O) at (0,0);
215 \coordinate (A) at (15:5);
216 \coordinate (B) at (15:9);
217 \coordinate (C) at (15:10);
218 \coordinate (X) at (15:14);
220 \coordinate (D) at (15:14.4);
221 \coordinate (E) at (15:-0.4);
223 \coordinate (A2) at (-7:4.5);
224 \coordinate (B2) at (-7:8.1);
225 \coordinate (C2) at (-7:9);
226 \coordinate (X2) at (-7:12.6);
228 \coordinate (D2) at (-7:13);
229 \coordinate (E2) at (-7:-0.4);
232 \draw (A) node[anchor=south east] {$A$};
233 \draw (A2) node[anchor=north east] {$A'$};
234 \draw (B) node[anchor=south east] {$B$};
235 \draw (B2) node[anchor=north east] {$B'$};
236 \draw (C) node[anchor=south east] {$C$};
237 \draw (C2) node[anchor=north east] {$C'$};
238 \draw (X) node[anchor=south east] {$D$};
239 \draw (X2) node[anchor=north east] {$D'$};
242 % Extenc les línies de A-A', ...
243 % pàgina 60 del manual de TikZ
244 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
245 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
247 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
248 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
250 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
251 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
253 \draw[thick, color=blue] (X) -- ($ (X) ! 1.1 ! (X2) $) coordinate (I);
254 \draw[thick, color=blue] (X2) -- ($ (X2) ! 1.1 ! (X) $) coordinate (I2);
257 % Dibuix les línies secants
258 \draw[very thick, color=blue] (E) -- (D);
259 \draw[very thick, color=blue] (E2) -- (D2);
261 % Dibuix els segments paral·lels
262 \draw[thick, color=blue] (F) -- (F2);
263 \draw[thick, color=blue] (G) -- (G2);
264 \draw[thick, color=blue] (H) -- (H2);
265 \draw[thick, color=blue] (I) -- (I2);
268 \foreach \point in {O,A,B,C,A2,B2,C2,X,X2}
269 \fill [black] (\point) circle (2pt);
278 \item $\overbar{A'C'} = 7,2 \unit{centi meter}$, $\overbar{OA} = 4,5 \unit{centi meter}$, $\overbar{CD} = 4,4 \unit{centi meter}$, $\overbar{B'D'} = 6,7 \unit{centi meter}$, $\overbar{AC} = 7,6 \unit{centi meter}$.
280 \startplacefigure[location=force, reference=fig:exercici-Tales-5, title={Figura e}]
281 \bTABLE[frame=off,align=middle]
286 % pàgina 58 del manual de TikZ
287 \coordinate[label=below:$O$] (O) at (0,0);
289 \coordinate (A) at (15:3);
290 \coordinate (B) at (15:7);
291 \coordinate (C) at (15:9);
292 \coordinate (X) at (15:12);
294 \coordinate (D) at (15:13.4);
295 \coordinate (E) at (15:-0.4);
297 \coordinate (A2) at (-7:3.75);
298 \coordinate (B2) at (-7:8.75);
299 \coordinate (C2) at (-7:11.25);
300 \coordinate (X2) at (-7:15);
302 \coordinate (D2) at (-7:15.4);
303 \coordinate (E2) at (-7:-0.4);
306 \draw (A) node[anchor=north east] {$A$};
307 \draw (A2) node[anchor=north east] {$A'$};
308 \draw (B) node[anchor=north east] {$B$};
309 \draw (B2) node[anchor=north east] {$B'$};
310 \draw (C) node[anchor=north east] {$C$};
311 \draw (C2) node[anchor=north east] {$C'$};
312 \draw (X) node[anchor=north east] {$D$};
313 \draw (X2) node[anchor=north east] {$D'$};
316 % Extenc les línies de A-A', ...
317 % pàgina 60 del manual de TikZ
318 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
319 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
321 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
322 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
324 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
325 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
327 \draw[thick, color=blue] (X) -- ($ (X) ! 1.1 ! (X2) $) coordinate (I);
328 \draw[thick, color=blue] (X2) -- ($ (X2) ! 1.1 ! (X) $) coordinate (I2);
331 % Dibuix les línies secants
332 \draw[very thick, color=blue] (E) -- (D);
333 \draw[very thick, color=blue] (E2) -- (D2);
335 % Dibuix els segments paral·lels
336 \draw[thick, color=blue] (F) -- (F2);
337 \draw[thick, color=blue] (G) -- (G2);
338 \draw[thick, color=blue] (H) -- (H2);
339 \draw[thick, color=blue] (I) -- (I2);
342 \foreach \point in {O,A,B,C,A2,B2,C2,X,X2}
343 \fill [black] (\point) circle (2pt);
352 \item $\overbar{BD} = 4,6 \unit{centi meter}$, $\overbar{AB} = 3,1 \unit{centi meter}$, $\overbar{B'D'} = 5,9 \unit{centi meter}$, $\overbar{OB'} = 9,5 \unit{centi meter}$, $\overbar{CD} = 2,4 \unit{centi meter}$.
354 \startplacefigure[location=force, reference=fig:exercici-Tales-6, title={Figura f}]
355 \bTABLE[frame=off,align=middle]
360 % pàgina 58 del manual de TikZ
361 \coordinate[label=below:$O$] (O) at (0,0);
363 \coordinate (A) at (15:4);
364 \coordinate (B) at (15:7);
365 \coordinate (C) at (15:9);
366 \coordinate (X) at (15:12);
368 \coordinate (D) at (15:12.4);
369 \coordinate (E) at (15:-0.4);
371 \coordinate (A2) at (-7:3.33);
372 \coordinate (B2) at (-7:5.83);
373 \coordinate (C2) at (-7:7.5);
374 \coordinate (X2) at (-7:10);
376 \coordinate (D2) at (-7:10.4);
377 \coordinate (E2) at (-7:-0.4);
380 \draw (A) node[anchor=south east] {$A$};
381 \draw (A2) node[anchor=north east] {$A'$};
382 \draw (B) node[anchor=south east] {$B$};
383 \draw (B2) node[anchor=north east] {$B'$};
384 \draw (C) node[anchor=south east] {$C$};
385 \draw (C2) node[anchor=north east] {$C'$};
386 \draw (X) node[anchor=south east] {$D$};
387 \draw (X2) node[anchor=north east] {$D'$};
390 % Extenc les línies de A-A', ...
391 % pàgina 60 del manual de TikZ
392 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
393 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
395 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
396 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
398 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
399 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
401 \draw[thick, color=blue] (X) -- ($ (X) ! 1.1 ! (X2) $) coordinate (I);
402 \draw[thick, color=blue] (X2) -- ($ (X2) ! 1.1 ! (X) $) coordinate (I2);
405 % Dibuix les línies secants
406 \draw[very thick, color=blue] (E) -- (D);
407 \draw[very thick, color=blue] (E2) -- (D2);
409 % Dibuix els segments paral·lels
410 \draw[thick, color=blue] (F) -- (F2);
411 \draw[thick, color=blue] (G) -- (G2);
412 \draw[thick, color=blue] (H) -- (H2);
413 \draw[thick, color=blue] (I) -- (I2);
416 \foreach \point in {O,A,B,C,A2,B2,C2,X,X2}
417 \fill [black] (\point) circle (2pt);
431 \startexercici[reference=exer:Tales-indirecta] Trobeu les dades que falten
433 \startplacefigure[location={force, none}, reference=fig:exercici-Tales-7, title={Exercici Tales 7}]
434 \bTABLE[frame=off,align=middle]
438 % punts de l'esquerra
439 \coordinate (A) at (0,0);
440 \coordinate (B) at (-95:1.8);
441 \coordinate (C) at (-95:2.7);
442 \coordinate (D) at (-95:4.7);
445 \startscope[shift={(5,0)}]
446 \coordinate (A2) at (-70:0);
447 \coordinate (B2) at (-70:1.8);
448 \coordinate (C2) at (-70:2.7);
449 \coordinate (D2) at (-70:4.7);
453 \draw (A) node[anchor=south east] {$A$};
454 \draw (A2) node[anchor=south west] {$A'$};
455 \draw (B) node[anchor=south east] {$B$};
456 \draw (B2) node[anchor=south west] {$B'$};
457 \draw (C) node[anchor=south east] {$C$};
458 \draw (C2) node[anchor=south west] {$C'$};
459 \draw (D) node[anchor=south east] {$D$};
460 \draw (D2) node[anchor=south west] {$D'$};
463 \draw ($ (A) ! 0.5 ! (B) $) node[anchor=west] {$18 \unit{milli meter}$};
464 \draw ($ (B) ! 0.5 ! (C) $) node[anchor=west] {$9 \unit{milli meter}$};
465 \draw ($ (C) ! 0.5 ! (D) $) node[anchor=west] {?};
467 \draw ($ (A2) ! 0.5 ! (B2) $) node[anchor=east] {$25 \unit{milli meter}$};
468 \draw ($ (B2) ! 0.5 ! (C2) $) node[anchor=east] {?};
469 \draw ($ (C2) ! 0.5 ! (D2) $) node[anchor=east] {$9 \unit{milli meter}$};
471 % Extenc les línies de A-A', ...
472 \draw[thick, color=blue] ($ (A) ! -0.2cm ! (A2) $) -- ($ (A2) ! -0.2cm ! (A) $);
473 \draw[thick, color=blue] ($ (B) ! -0.2cm ! (B2) $) -- ($ (B2) ! -0.2cm ! (B) $);
474 \draw[thick, color=blue] ($ (C) ! -0.2cm ! (C2) $) -- ($ (C2) ! -0.2cm ! (C) $);
475 \draw[thick, color=blue] ($ (D) ! -0.2cm ! (D2) $) -- ($ (D2) ! -0.2cm ! (D) $);
476 \draw[thick, color=blue] ($ (A) ! -0.2cm ! (D) $) -- ($ (D) ! -0.2cm ! (A) $);
477 \draw[thick, color=blue] ($ (A2) ! -0.2cm ! (D2) $) -- ($ (D2) ! -0.2cm ! (A2) $);
480 \foreach \point in {O,A,B,C,A2,B2,C2,D,D2}
481 \fill [black] (\point) circle (2pt);
491 \startexercici Calculeu l'altura d'una muntanya a partir de les dades següents: Des d'un vaixell es calcula que la distància del cim de la muntanya és de 4525 m, amb un angle $\alpha$ que forma la visual sobre l'horitzó. Sobre un paper es dibuixa un triangle rectangle $\triangle ABC$, que té $\alpha$ com un dels angles aguts. El catet $\overbar{BC}$ mesura $21 \unit{centi meter}$ i la hipotenusa $\overbar{AC}$, $83 \unit{centi meter}$.
494 \startexercici Trobeu la longitud dels costats que falten:
498 \item $\overbar{AB} = 2$, $\overbar{BC} = 3$, $\overbar{B'C'} = 6$.
499 \startplacefigure[location={force, none}, reference=fig:exercici-triangles-semblants-1, title={Teorema de Tales amb trignales semblants}]
500 \bTABLE[frame=off,align=middle]
502 \bTD[align={middle,lohi}] {\starttikzpicture[scale=1]
504 \startscope[shift={(0,0)}]
505 \filldraw[fill=green!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
506 \draw (0,0) node[anchor=north] {$A$};
507 \draw (2,1) node[anchor=north] {$B$};
508 \draw (1,3) node[anchor=south] {$C$};
513 \bTD[align=middle] {\starttikzpicture[scale=2]
515 \startscope[shift={(0,0)}]
516 \filldraw[fill=red!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
517 \draw (0,0) node[anchor=north] {$A'$};
518 \draw (2,1) node[anchor=north] {$B'$};
519 \draw (1,3) node[anchor=south] {$C'$};
528 \item $\overbar{A'C'} = 2$, $\overbar{AC} = 3$, $\overbar{A'B'} = 3$.
529 \startplacefigure[location={force,none}, reference=fig:exercici-triangles-semblants-2, title={Teorema de Tales amb trignales semblants}]
530 \bTABLE[frame=off,align=middle]
532 \bTD[align={middle,lohi}] {\starttikzpicture[scale=1]
534 \startscope[shift={(0,0)}]
535 \filldraw[fill=green!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
536 \draw (0,0) node[anchor=north] {$A$};
537 \draw (2,1) node[anchor=north] {$B$};
538 \draw (1,3) node[anchor=south] {$C$};
543 \bTD[align=middle] {\starttikzpicture[scale=2]
545 \startscope[shift={(0,0)}]
546 \filldraw[fill=red!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
547 \draw (0,0) node[anchor=north] {$A'$};
548 \draw (2,1) node[anchor=north] {$B'$};
549 \draw (1,3) node[anchor=south] {$C'$};
558 \item $\overbar{AB} = 2$, $\overbar{BC} = 3$, $\overbar{B'C'} = 6$.
559 \startplacefigure[location={force, none}, reference=fig:exercici-triangles-semblants-3, title={Teorema de Tales amb trignales semblants}]
560 \bTABLE[frame=off,align=middle]
562 \bTD[align={middle,lohi}] {\starttikzpicture[scale=1.5,rotate around={45:(0,0)}]
564 \startscope[shift={(0,0)}]
565 \filldraw[fill=green!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
566 \draw (0,0) node[anchor=north] {$A$};
567 \draw (2,1) node[anchor=north west] {$B$};
568 \draw (1,3) node[anchor=south] {$C$};
569 \filldraw[fill=red!50,very thick] (0, 0) -- (1, 0.5) -- (0.5, 1.5) -- cycle;
570 \draw (1,0.5) node[anchor=north west] {$B'$};
571 \draw (0.5,1.5) node[anchor=east] {$C'$};
580 \item $\overbar{AC} = 3$, $\overbar{AB} = 1$, $\overbar{BC} = 5$, $\overbar{B'C'} = 10$.
581 \startplacefigure[location={force, none}, reference=fig:exercici-triangles-semblants-4, title={Teorema de Tales amb trignales semblants}]
582 \bTABLE[frame=off,align=middle]
584 \bTD[align={middle,lohi}] {\starttikzpicture[scale=2]
586 \startscope[shift={(0,0)}]
587 \filldraw[fill=green!50,very thick] (0, 0) -- (2, 1) -- (1, 3) -- cycle;
588 \draw (0,0) node[anchor=north] {$A$};
589 \draw (2,1) node[anchor=north] {$B$};
590 \draw (1,3) node[anchor=south] {$C$};
591 \filldraw[fill=red!50,very thick] (0, 0) -- (1, 0.5) -- (0.5, 1.5) -- cycle;
592 \draw (1,0.5) node[anchor=north] {$B'$};
593 \draw (0.5,1.5) node[anchor=east] {$C'$};
606 \startexercici[reference=exer:exercici-Tales-meu] Trobeu les mesures dels costats que falten.
610 \item $\overbar{OA} = 7,2$, $\overbar{OA'} = 5,34$, $\overbar{OB} = 12$, $\overbar{D'C'} = 3.56$
612 \startplacefigure[location=force, reference=fig:exercici-Tales-9, title={Figura a}]
613 \bTABLE[frame=off,align=middle]
618 % pàgina 58 del manual de TikZ
619 \coordinate[label=below:$O$] (O2) at (0:0);
620 \coordinate (C) at (12:7);
621 \coordinate (C2) at (-10:8.90);
622 \coordinate (CL) at (192:7);
623 \coordinate (C2L) at (170:8.90);
625 \coordinate (AR) at ($ (O2) ! 0.6 ! (C) $);
626 \coordinate (AL) at ($ (O2) ! 0.6 ! (CL) $);
628 \coordinate (A2R) at ($ (O2) ! 0.6 ! (C2) $);
629 \coordinate (A2L) at ($ (O2) ! 0.6 ! (C2L) $);
631 % Dibuix les línies secants
632 \draw[very thick, color=blue] ($ (C) ! -0.4cm ! (O2)$) -- (O2);
633 \draw[very thick, color=blue] ($ (C2) ! -0.4cm ! (O2)$) -- (O2);
634 \draw[very thick, color=blue] ($ (AR) ! -0.2cm ! (A2R)$) -- ($ (A2R) ! -0.2cm ! (AR)$);
635 \draw[very thick, color=blue] ($ (C) ! -0.2cm ! (C2)$) -- ($ (C2) ! -0.2cm ! (C)$);
637 \draw[very thick, color=blue] ($ (CL) ! -0.4cm ! (O2)$) -- (O2);
638 \draw[very thick, color=blue] ($ (C2L) ! -0.4cm ! (O2)$) -- (O2);
639 \draw[very thick, color=blue] ($ (AL) ! -0.2cm ! (A2L)$) -- ($ (A2L) ! -0.2cm ! (AL)$);
640 \draw[very thick, color=blue] ($ (CL) ! -0.2cm ! (C2L)$) -- ($ (C2L) ! -0.2cm ! (CL)$);
645 \draw (AR) node[anchor=south west] {$A$};
646 \draw (A2R) node[anchor=north east] {$A'$};
647 \draw (C) node[anchor=south west] {$B$};
648 \draw (C2) node[anchor=north east] {$B'$};
650 \draw (AL) node[anchor=north west] {$C$};
651 \draw (A2L) node[anchor=south west] {$C'$};
652 \draw (CL) node[anchor=north west] {$D$};
653 \draw (C2L) node[anchor=south west] {$D'$};
656 \foreach \point in {O2,AR,A2R,C,C2,AL,A2L,CL,C2L}
657 \fill [black] (\point) circle (2pt);
665 \item $\overbar{OA} = 4$, $\overbar{OA''} = 5$, $\overbar{A'B'} = 2,25$, $\overbar{A''C''} = 6,25$, $\overbar{BC} = 2$, $\overbar{C'D'} = 2,25$
666 \startplacefigure[location=force, reference=fig:exercici-Tales-7, title={Figura b}]
667 \bTABLE[frame=off,align=middle]
672 % pàgina 58 del manual de TikZ
673 \coordinate[label=below:$O$] (O) at (0,0);
675 \coordinate (A) at (15:4);
676 \coordinate (B) at (15:7);
677 \coordinate (C) at (15:9);
678 \coordinate (X) at (15:12);
680 \coordinate (D) at (15:12.4);
681 \coordinate (E) at (15:-0.4);
683 \coordinate (A2) at (-7:3);
684 \coordinate (B2) at (-7:5.25);
685 \coordinate (C2) at (-7:6.75);
686 \coordinate (X2) at (-7:9);
688 \coordinate (D2) at (-7:9.4);
689 \coordinate (E2) at (-7:-0.4);
691 \coordinate (A3) at (-20:5);
692 \coordinate (B3) at (-20:8.75);
693 \coordinate (C3) at (-20:11.25);
694 \coordinate (X3) at (-20:15);
696 \coordinate (D3) at (-20:15.4);
697 \coordinate (E3) at (-20:-0.4);
701 \draw (A) node[anchor=south east] {$A$};
702 \draw (A2) node[anchor=north east] {$A'$};
703 \draw (A3) node[anchor=north east] {$A''$};
704 \draw (B) node[anchor=south east] {$B$};
705 \draw (B2) node[anchor=north east] {$B'$};
706 \draw (B3) node[anchor=north east] {$B''$};
707 \draw (C) node[anchor=south east] {$C$};
708 \draw (C2) node[anchor=north east] {$C'$};
709 \draw (C3) node[anchor=north east] {$C''$};
710 \draw (X) node[anchor=south east] {$D$};
711 \draw (X2) node[anchor=north east] {$D'$};
712 \draw (X3) node[anchor=north east] {$D''$};
715 % Extenc les línies de A-A', ...
716 % pàgina 60 del manual de TikZ
717 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
718 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
720 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
721 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
723 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
724 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
726 \draw[thick, color=blue] (X) -- ($ (X) ! 1.1 ! (X2) $) coordinate (I);
727 \draw[thick, color=blue] (X2) -- ($ (X2) ! 1.1 ! (X) $) coordinate (I2);
729 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A3) $) coordinate (F23);
730 \draw[thick, color=blue] (A3) -- ($ (A3) ! 1.2 ! (A2) $) coordinate (F32);
732 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B3) $) coordinate (G23);
733 \draw[thick, color=blue] (B3) -- ($ (B3) ! 1.1 ! (B2) $) coordinate (G32);
735 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C3) $) coordinate (H23);
736 \draw[thick, color=blue] (C3) -- ($ (C3) ! 1.1 ! (C2) $) coordinate (H32);
738 \draw[thick, color=blue] (X2) -- ($ (X2) ! 1.1 ! (X3) $) coordinate (I23);
739 \draw[thick, color=blue] (X3) -- ($ (X3) ! 1.1 ! (X2) $) coordinate (I32);
742 % Dibuix les línies secants
743 \draw[very thick, color=blue] (E) -- (D);
744 \draw[very thick, color=blue] (E2) -- (D2);
745 \draw[very thick, color=blue] (E3) -- (D3);
747 % Dibuix els segments paral·lels
748 \draw[thick, color=green!70!black] (F) -- (F2);
749 \draw[thick, color=green!70!black] (G) -- (G2);
750 \draw[thick, color=green!70!black] (H) -- (H2);
751 \draw[thick, color=green!70!black] (I) -- (I2);
752 \draw[thick, color=red!70!black] (F23) -- (F32);
753 \draw[thick, color=red!70!black] (G23) -- (G32);
754 \draw[thick, color=red!70!black] (H23) -- (H32);
755 \draw[thick, color=red!70!black] (I23) -- (I32);
758 \foreach \point in {O,A,B,C,A2,B2,C2,A3,B3,C3,X,X2,X3}
759 \fill [black] (\point) circle (2pt);
768 \item $\overbar{OA} = 4,2$, $\overbar{AB} = 3$, $\overbar{A'B'} = 2,43$, $\overbar{B'C'} = 3,07$, $\overbar{CD} = 1,2$, $\overbar{DO'} = 4,8$, $\overbar{D'O'} = 2,07$, $\overbar{D'E'} = 1,25$, $\overbar{EO'} = 0,92$
770 \startplacefigure[location=force, reference=fig:exercici-Tales-8, title={Figura c}]
771 \bTABLE[frame=off,align=middle]
776 % pàgina 58 del manual de TikZ
777 \coordinate[label=below:$O$] (O) at (0,0);
779 \coordinate (A) at (20:4.2);
780 \coordinate (B) at (20:7.2);
781 \coordinate (C) at (20:11);
783 \coordinate (D) at (20:11.4);
784 \coordinate (E) at (20:-0.4);
786 \coordinate (A2) at (-10:3.4);
787 \coordinate (B2) at (-10:5.83);
788 \coordinate (C2) at (-10:8.90);
790 \coordinate (D2) at (-10:9.3);
791 \coordinate (E2) at (-10:-0.4);
794 \draw (A) node[anchor=south east] {$A$};
795 \draw (A2) node[anchor=north east] {$A'$};
796 \draw (B) node[anchor=south east] {$B$};
797 \draw (B2) node[anchor=north east] {$B'$};
798 \draw (C) node[anchor=south east] {$C$};
799 \draw (C2) node[anchor=north east] {$C'$};
802 % Extenc les línies de A-A', ...
803 % pàgina 60 del manual de TikZ
804 \draw[thick, color=blue] (A) -- ($ (A) ! 1.2 ! (A2) $) coordinate (F);
805 \draw[thick, color=blue] (A2) -- ($ (A2) ! 1.2 ! (A) $) coordinate (F2);
807 \draw[thick, color=blue] (B) -- ($ (B) ! 1.1 ! (B2) $) coordinate (G);
808 \draw[thick, color=blue] (B2) -- ($ (B2) ! 1.1 ! (B) $) coordinate (G2);
810 \draw[thick, color=blue] (C) -- ($ (C) ! 1.1 ! (C2) $) coordinate (H);
811 \draw[thick, color=blue] (C2) -- ($ (C2) ! 1.1 ! (C) $) coordinate (H2);
813 % Dibuix les línies secants
814 \draw[very thick, color=blue] (E) -- (D);
815 \draw[very thick, color=blue] (E2) -- (D2);
817 % Dibuix els segments paral·lels
818 \draw[thick, color=blue] (F) -- (F2);
819 \draw[thick, color=blue] (G) -- (G2);
820 \draw[thick, color=blue] (H) -- (H2);
823 \foreach \point in {O,A,B,C,A2,B2,C2}
824 \fill [black] (\point) circle (2pt);
827 % pàgina 58 del manual de TikZ
828 \coordinate[label=below:$O'$] (O2) at (0:15);
830 \coordinate (AR) at ($ (O2) ! 0.8 ! (C) $);
831 \coordinate (BR) at ($ (O2) ! 0.4 ! (AR) $);
833 \coordinate (A2R) at ($ (O2) ! 0.7 ! (C2) $);
834 \coordinate (B2R) at ($ (O2) ! 0.4 ! (A2R) $);
836 % Dibuix les línies secants
837 \draw[very thick, color=orange!80!black] (C) -- ($ (O2) ! -0.4cm ! (C)$);
838 \draw[very thick, color=orange!80!black] (C2) -- ($ (O2) ! -0.4cm ! (C2)$);
839 \draw[very thick, color=orange!80!black] ($ (AR) ! -0.4cm ! (A2R)$) -- ($ (A2R) ! -0.4cm ! (AR)$);
840 \draw[very thick, color=orange!80!black] ($ (BR) ! -0.4cm ! (B2R)$) -- ($ (B2R) ! -0.4cm ! (BR)$);
843 \draw (AR) node[anchor=south west] {$D$};
844 \draw (A2R) node[anchor=north west] {$D'$};
845 \draw (BR) node[anchor=south west] {$E$};
846 \draw (B2R) node[anchor=north west] {$E'$};
850 \foreach \point in {O2,AR,BR,A2R,B2R}
851 \fill [black] (\point) circle (2pt);
864 \startexercici Calculeu l'altura d'un xiprer que projecta una ombra de 5 metres sabent que, en el mateix moment, un castanyer petit, d'1,8 \unit{meter}\space d'altura, projecta una ombra de 60 \unit{centi meter}.
867 \startexercici Què fa l'altura d'una torre que projecta una ombra de 18,5 \unit{meter}\space si sabem que, en el mateix moment, un arbre de 3,5 \unit{meter}\space d'altura projecta una ombra de 0,75 \unit{meter}.
870 \startexercici[reference=exer:la-escala, title={L'escala}] Volem construir una escala triangular, a partir de dos pals de fusta de 2 metres de llarg cadascun (figura~\in[fig:exercici-escala]). L'únic que sabem és que la separació entre els dos pals és de 1,5 metres a la base i que tendrà escalons cada $30 \unit{centi meter}$. Quina amplada tendran aquests escalons?
872 \startplacefigure[location=force, reference=fig:exercici-escala, title={Esquema de l'escala triangular}]
873 \bTABLE[frame=off,align=middle]
877 \coordinate (A) at (0,0);
878 \coordinate (A2) at (0.15, 0);
879 \coordinate (A3) at (0.25, 0);
880 \coordinate (A4) at (-0.35, 0);
881 \coordinate (B) at (2.25,0);
882 \coordinate (B2) at (2.1,0);
883 \coordinate (B3) at (2,0);
884 \coordinate (C) at (1.125,2.775);
885 \coordinate (C2) at (1.125, 2.55);
886 \coordinate (C3) at (0.775,2.775);
887 \filldraw[pattern=crosshatch, pattern color=orange!50!black] (A) -- (C) -- (C2) -- (A2) -- cycle;
888 \filldraw[pattern=crosshatch, pattern color=orange!50!black] (B) -- (C) -- (C2) -- (B2) -- cycle;
889 \draw[<->] (A3) -- (B3);
890 \draw (1.125, 0) node[fill=white] {$1,5$};
891 \draw[<->] (A4) -- (C3);
892 \draw ($ (A4) ! 0.5 ! (C3) $) node[fill=white] {$2$};
900 \startexercici[title={Tallar el pastís}] Tenim un triangle equilàter de costat $10 \unit{centi meter}$. El volem tallar amb un sol tall recte paral·lel a la seva base (figura~\in[figura:problemes-2]). A quina distància de la base hem de fer el tall per a què les àrees resultants siguin iguals?
902 \startplacefigure[location=force, reference=figura:problemes-2, title={Tall paral·lel a la base}]
906 \starttikzpicture[scale=0.6]
907 \node[regular polygon, regular polygon sides=3, minimum size=5cm, draw] at (0,0) {};
908 \draw[color=blue!80, loosely dashed] (-4,1) -- (4,1);
920 \startsubject[title={Solucions}]
922 \startitemize[1][distance=0.5cm]
923 \sym{\in[exercici:Tales-directe]}
924 \startitemize[a, packed]
925 \item $\overbar{BC} = 4,7 \unit{centi meter}$, $\overbar{A'B'} \simeq 2,43 \unit{centi meter}$, $\overbar{B'C'} = 3,80 \unit{centi meter}$
926 \item $\overbar{AB} = 3,1 \unit{centi meter}$, $\overbar{BC} = 4,5 \unit{centi meter}$, $\overbar{OA'} \simeq 4,91 \unit{centi meter}$, $\overbar{A'B'} \simeq 3,24 \unit{centi meter}$
927 \item $\overbar{AB} = 7,74 \unit{centi meter}$, $\overbar{BC} = 3,96 \unit{centi meter}$
928 \item $\overbar{OA'} \simeq 6,24 \unit{centi meter}$, $\overbar{A'B'} \simeq 4,77 \unit{centi meter}$, $\overbar{B'C'} \simeq 3,33 \unit{centi meter}$, $\overbar{C'D'} \simeq 3,77 \unit{centi meter}$, $\overbar{BC} \simeq 2,72 \unit{centi meter}$, $\overbar{CD} \simeq 3,08 \unit{centi meter}$
929 \item $\overbar{OA'} = 3,08 \unit{centi meter}$, $\overbar{C'D'} \simeq 4,17 \unit{centi meter}$, $\overbar{B'C'} \simeq 2,53 \unit{centi meter}$, $\overbar{A'B'} \simeq 4,67 \unit{centi meter}$, $\overbar{AB} \simeq 4,93 \unit{centi meter}$, $\overbar{BC} \simeq 2,67 \unit{centi meter}$
930 \item $\overbar{BC} = 2,2 \unit{centi meter}$, $\overbar{OA} \simeq 4,31 \unit{centi meter}$, $\overbar{OA'} = 5,53 \unit{centi meter}$, $\overbar{B'C'} = 2,82 \unit{centi meter}$, $\overbar{A'B'} \simeq 3,97 \unit{centi meter}$, $\overbar{C'D'} \simeq 3,08 \unit{centi meter}$
933 \sym{\in[exer:exercici-Tales-meu]}
934 \startitemize[a,packed]
935 \item $\overbar{OC'}=5,34$, $\overbar{CD}=2,8$, $\overbar{OC}=4,2$, $\overbar{AB}=4,8$, $\overbar{A'B'}=3,56$
936 \item $\overbar{OA'} = 3$, $\overbar{AB} = 3$, $\overbar{A''B''} = 3,75$, $\overbar{B'C'} = 1,5$, $\overbar{B''C''} = 2,5$, $\overbar{CD} = 3$, $\overbar{C''D''} = 3,75$
937 \item $\overbar{OA'} = 3,4$, $\overbar{BC} = 3,8$, $\overbar{C'D'} = 0,89$, $\overbar{DE} = 3,88$, $\overbar{E'O} = 0,82$
940 \sym{\in[exer:la-escala]} L'alçada total de l'escala és d'aproximadament 1,85 m