Rotating 90°

<p><span style="font-size:24px"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-rotating-90" target="_blank">Rotating 90&deg; Exercise</a></span></p><p>Whenever you fix a line segment at the origin on the $xy$-coordinate plane, and then <span style="color:#8e44ad;">Rotate the non-fixed Point 90&deg;</span>, what happens?</p> <center><svg height="229.22857666015625" style=" width:661.8285522460938px; height:229.22857666015625px; background: #FFF; fill: none; " width="661.8285522460938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape axis2d" style="stroke-width: 1; stroke: rgb(0, 0, 0);"><path class="real" d=" M173,167.74 L479.6,167.74 M331.21,11 L331.21,199.26"></path><path d=" M472.6,162.74 L479.6,167.74 L472.6,172.74"></path><path d=" M326.21,18 L331.21,11 L336.21,18"></path></g><g class="arrow-line"><path class="connection real" d=" M423.6,108.26 L331.21,167.74" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M421.23,108.26 C421.23,106.95 422.29,105.89 423.6,105.89 C424.91,105.89 425.97,106.95 425.97,108.26 C425.97,109.57 424.91,110.63 423.6,110.63 C422.29,110.63 421.23,109.57 421.23,108.26 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M266.2,74.4 L331.21,167.74" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M263.83,74.4 C263.83,73.09 264.89,72.03 266.2,72.03 C267.51,72.03 268.57,73.09 268.57,74.4 C268.57,75.71 267.51,76.77 266.2,76.77 C264.89,76.77 263.83,75.71 263.83,74.4 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M324.84,159.31 L333.27,152.95 L339.63,161.37" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="227.39999389648438" style="width:660px;height:227.39999389648438px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><g transform="matrix(1,0,0,1,426.50714111328125,126.17141723632812)"><path d="M146 266C146 526 243 632 301 700L282 726C225 675 60 542 60 266C60 159 85 58 133 -32C168 -99 200 -138 282 -215L301 -194C255 -137 146 -15 146 266Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,432.14996337890625,126.17141723632812)"><path d="M462 224C462 345 355 366 308 374C388 436 418 482 418 541C418 630 344 689 233 689C165 689 120 670 72 622L43 498L74 498L92 554C103 588 166 622 218 622C283 622 336 569 336 506C336 431 277 368 206 368C198 368 187 369 174 370L159 371L147 318L154 312C192 329 211 334 238 334C321 334 369 281 369 190C369 88 308 21 215 21C169 21 128 36 98 64C74 86 61 109 42 163L15 153C36 92 44 56 50 6C103 -12 147 -20 184 -20C307 -20 462 87 462 224Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,440.63568115234375,126.17141723632812)"><path d="M204 123C177 114 159 108 106 93C99 17 74 -48 16 -144L30 -155L71 -136C152 -31 190 32 218 109Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,448.27142333984375,126.17141723632812)"><path d="M265 23L265 -3C452 -3 452 0 488 0C524 0 524 -3 717 -3L717 82C602 77 556 81 371 77L553 270C650 373 680 428 680 503C680 618 602 689 475 689C403 689 354 669 305 619L288 483L317 483L330 529C346 587 382 612 449 612C535 612 590 558 590 473C590 398 548 324 435 204Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,460.9857177734375,126.17141723632812)"><path d="M51 726L32 700C87 636 187 526 187 266C187 -10 83 -131 32 -194L51 -215C104 -165 273 -23 273 265C273 542 108 675 51 726Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,222.50714111328125,92.17141723632812)"><path d="M146 266C146 526 243 632 301 700L282 726C225 675 60 542 60 266C60 159 85 58 133 -32C168 -99 200 -138 282 -215L301 -194C255 -137 146 -15 146 266Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,228.14996337890625,92.17141723632812)"><path d="M148 254C148 224 162 200 192 181L213 195C197 214 191 227 191 243C191 276 204 293 285 366C370 444 395 483 395 540C395 627 316 694 213 694C169 694 134 685 81 659L43 546L72 538L92 586C103 613 134 628 179 628C251 628 302 583 302 519C302 416 148 344 148 254ZM198 111C168 111 141 83 141 53C141 23 168 -5 197 -5C229 -5 257 22 257 53C257 83 229 111 198 111ZM647 123C620 114 602 108 549 93C542 17 517 -48 459 -144L473 -155L514 -136C595 -31 633 32 661 109Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,243.31427001953125,92.17141723632812)"><path d="" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,247.54998779296875,92.17141723632812)"><path d="M148 254C148 224 162 200 192 181L213 195C197 214 191 227 191 243C191 276 204 293 285 366C370 444 395 483 395 540C395 627 316 694 213 694C169 694 134 685 81 659L43 546L72 538L92 586C103 613 134 628 179 628C251 628 302 583 302 519C302 416 148 344 148 254ZM198 111C168 111 141 83 141 53C141 23 168 -5 197 -5C229 -5 257 22 257 53C257 83 229 111 198 111Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,258.4857177734375,92.17141723632812)"><path d="M51 726L32 700C87 636 187 526 187 266C187 -10 83 -131 32 -194L51 -215C104 -165 273 -23 273 265C273 542 108 675 51 726Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,222.91424560546875,60.914276123046875)"><path d="M167 420C188 420 201 421 226 424C171 350 72 266 72 126C72 44 107 -11 160 -11C224 -11 276 44 350 187C341 143 338 118 338 95C338 32 367 -11 412 -11C546 -11 699 238 699 384C699 441 673 482 637 482C624 482 612 478 604 470L554 423L565 404C574 410 583 413 592 413C619 413 635 390 635 350C635 205 538 39 453 39C424 39 403 79 403 134C403 181 412 247 424 290L476 474L469 482L395 460L366 311C354 249 340 209 311 155C271 80 233 39 203 39C168 39 145 78 145 136C145 255 186 352 285 464L275 482C255 473 242 470 214 470C186 470 144 472 115 475L103 476C97 477 92 477 91 477C79 477 70 475 59 470C45 443 34 407 21 352L43 352L60 394C67 411 87 422 111 422C145 422 128 420 167 420ZM992 204L963 77C959 60 957 42 957 26C957 4 966 -9 981 -9C1004 -9 1045 17 1127 85L1120 106C1096 86 1067 59 1045 59C1036 59 1030 68 1030 82C1030 87 1030 90 1031 93L1123 472L1113 481L1080 463C1039 478 1022 482 995 482C967 482 947 477 920 464C858 433 825 403 800 354C756 265 725 145 725 67C725 23 740 -11 759 -11C796 -11 876 41 992 204ZM1040 414C1018 305 999 253 965 201C908 117 847 59 815 59C803 59 797 72 797 99C797 163 825 280 860 360C884 415 907 433 955 433C978 433 996 429 1040 414ZM1289 390L1233 107C1232 99 1220 61 1220 31C1220 6 1231 -9 1250 -9C1285 -9 1320 11 1398 74L1429 99L1419 117L1374 86C1345 66 1325 56 1314 56C1305 56 1300 64 1300 76C1300 102 1314 183 1343 328L1356 390L1463 390L1474 440C1436 436 1402 434 1364 434C1380 528 1391 577 1409 631L1398 646C1378 634 1351 622 1320 610L1295 440C1251 419 1225 408 1207 403L1205 390ZM2228 722L2216 733C2164 707 2128 698 2056 691L2052 670L2100 670C2124 670 2134 663 2134 646C2134 638 2133 629 2132 622L2104 468C2074 477 2047 482 2022 482C1953 482 1859 410 1813 323C1782 265 1762 170 1762 86C1762 21 1777 -11 1806 -11C1833 -11 1870 6 1904 33C1958 77 1991 116 2058 217L2035 126C2024 82 2019 50 2019 24C2019 3 2028 -9 2045 -9C2062 -9 2086 3 2120 28L2201 88L2191 107L2147 76C2133 66 2117 59 2108 59C2100 59 2094 68 2094 82C2094 90 2095 99 2102 128ZM1859 59C1843 59 1834 73 1834 98C1834 224 1880 380 1929 418C1942 428 1961 433 1988 433C2032 433 2061 427 2094 410L2082 351C2045 171 1904 59 1859 59ZM2515 204L2486 77C2482 60 2480 42 2480 26C2480 4 2489 -9 2504 -9C2527 -9 2568 17 2650 85L2643 106C2619 86 2590 59 2568 59C2559 59 2553 68 2553 82C2553 87 2553 90 2554 93L2646 472L2636 481L2603 463C2562 478 2545 482 2518 482C2490 482 2470 477 2443 464C2381 433 2348 403 2323 354C2279 265 2248 145 2248 67C2248 23 2263 -11 2282 -11C2319 -11 2399 41 2515 204ZM2563 414C2541 305 2522 253 2488 201C2431 117 2370 59 2338 59C2326 59 2320 72 2320 99C2320 163 2348 280 2383 360C2407 415 2430 433 2478 433C2501 433 2519 429 2563 414ZM3172 722L3160 733C3115 711 3074 697 3000 691L2996 670L3044 670C3062 670 3078 667 3078 647C3078 641 3078 632 3076 622L3034 388C3014 272 2972 80 2946 2L2953 -9L3022 7C3030 64 3044 164 3084 236C3129 317 3232 414 3274 414C3285 414 3296 407 3296 393C3296 375 3291 342 3281 303L3230 107C3224 85 3217 55 3217 31C3217 6 3227 -9 3248 -9C3280 -9 3348 41 3407 85L3397 103L3371 86C3348 71 3322 56 3310 56C3303 56 3297 65 3297 76C3297 88 3300 101 3304 116L3368 372C3374 398 3379 423 3379 447C3379 464 3373 482 3347 482C3312 482 3235 437 3167 374C3134 343 3108 308 3080 273L3076 275ZM3763 111L3739 94C3686 56 3638 36 3602 36C3555 36 3526 73 3526 133C3526 158 3529 185 3534 214C3551 218 3660 248 3685 259C3770 296 3809 342 3809 404C3809 451 3775 482 3725 482C3657 496 3547 423 3510 349C3480 299 3450 180 3450 113C3450 35 3494 -11 3566 -11C3623 -11 3679 17 3771 92ZM3548 274C3565 343 3585 386 3614 412C3632 428 3663 440 3687 440C3716 440 3735 420 3735 388C3735 344 3700 297 3648 272C3620 258 3584 247 3539 237ZM4074 722L4062 733C4010 707 3974 698 3902 691L3898 670L3946 670C3970 670 3980 663 3980 648C3980 645 3980 640 3977 622C3966 567 3893 182 3878 132C3865 82 3859 52 3859 31C3859 6 3870 -9 3889 -9C3915 -9 3951 12 4049 85L4039 103L4013 86C3984 67 3962 56 3952 56C3945 56 3939 66 3939 76C3939 82 3940 89 3943 104ZM4351 722L4339 733C4287 707 4251 698 4179 691L4175 670L4223 670C4247 670 4257 663 4257 648C4257 645 4257 640 4254 622C4243 567 4170 182 4155 132C4142 82 4136 52 4136 31C4136 6 4147 -9 4166 -9C4192 -9 4228 12 4326 85L4316 103L4290 86C4261 67 4239 56 4229 56C4222 56 4216 66 4216 76C4216 82 4217 89 4220 104Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,297.31427001953125,60.914276123046875)"><path d="M148 254C148 224 162 200 192 181L213 195C197 214 191 227 191 243C191 276 204 293 285 366C370 444 395 483 395 540C395 627 316 694 213 694C169 694 134 685 81 659L43 546L72 538L92 586C103 613 134 628 179 628C251 628 302 583 302 519C302 416 148 344 148 254ZM198 111C168 111 141 83 141 53C141 23 168 -5 197 -5C229 -5 257 22 257 53C257 83 229 111 198 111Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g></svg> </svg></center> <p>A lot of people make the mistake of thinking that the new point will be $(-3, 2)$, that all they had to do was change the sign of the $x$-coordinate because the line segment is now in Quadrant II. But actually this <strong><u>does NOT work</u></strong>.&nbsp;</p> <p><strong><span style="color:#000000;">Why?</span></strong></p> <p>Remember that the slopes of perpendicular lines are negative reciprocals of each other. If the new point equaled $(-3,2)$, its slope can be calculated with the origin point $(0,0)$ as $-\frac{2}{3}$. The slope of the original line is $\frac{2}{3}$. These are NOT negative reciprocals of each other.&nbsp;</p> <p><strong><span style="color:#27ae60;">The Correct Answer</span></strong></p> <p>Not only do you have to pay attention to what quadrant you&#39;re in to determine whether the $x$-coordinate, $y$-coordinate, or both are positive or negative, you also have to <span style="color:#e74c3c;">FLIP</span> the coordinates. So the mystery coordinate is actually...</p> <center><svg height="229.22857666015625" style=" width:661.8285522460938px; height:229.22857666015625px; background: #FFF; fill: none; " width="661.8285522460938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape axis2d" style="stroke-width: 1; stroke: rgb(0, 0, 0);"><path class="real" d=" M173,167.74 L479.6,167.74 M331.21,11 L331.21,199.26"></path><path d=" M472.6,162.74 L479.6,167.74 L472.6,172.74"></path><path d=" M326.21,18 L331.21,11 L336.21,18"></path></g><g class="arrow-line"><path class="connection real" d=" M423.6,108.26 L331.21,167.74" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M421.23,108.26 C421.23,106.95 422.29,105.89 423.6,105.89 C424.91,105.89 425.97,106.95 425.97,108.26 C425.97,109.57 424.91,110.63 423.6,110.63 C422.29,110.63 421.23,109.57 421.23,108.26 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M266.2,74.4 L331.21,167.74" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M263.83,74.4 C263.83,73.09 264.89,72.03 266.2,72.03 C267.51,72.03 268.57,73.09 268.57,74.4 C268.57,75.71 267.51,76.77 266.2,76.77 C264.89,76.77 263.83,75.71 263.83,74.4 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M324.84,159.31 L333.27,152.95 L339.63,161.37" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="227.39999389648438" style="width:660px;height:227.39999389648438px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><g transform="matrix(1,0,0,1,426.50714111328125,126.17141723632812)"><path d="M146 266C146 526 243 632 301 700L282 726C225 675 60 542 60 266C60 159 85 58 133 -32C168 -99 200 -138 282 -215L301 -194C255 -137 146 -15 146 266Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,432.14996337890625,126.17141723632812)"><path d="M462 224C462 345 355 366 308 374C388 436 418 482 418 541C418 630 344 689 233 689C165 689 120 670 72 622L43 498L74 498L92 554C103 588 166 622 218 622C283 622 336 569 336 506C336 431 277 368 206 368C198 368 187 369 174 370L159 371L147 318L154 312C192 329 211 334 238 334C321 334 369 281 369 190C369 88 308 21 215 21C169 21 128 36 98 64C74 86 61 109 42 163L15 153C36 92 44 56 50 6C103 -12 147 -20 184 -20C307 -20 462 87 462 224Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,440.63568115234375,126.17141723632812)"><path d="M204 123C177 114 159 108 106 93C99 17 74 -48 16 -144L30 -155L71 -136C152 -31 190 32 218 109Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,448.27142333984375,126.17141723632812)"><path d="M265 23L265 -3C452 -3 452 0 488 0C524 0 524 -3 717 -3L717 82C602 77 556 81 371 77L553 270C650 373 680 428 680 503C680 618 602 689 475 689C403 689 354 669 305 619L288 483L317 483L330 529C346 587 382 612 449 612C535 612 590 558 590 473C590 398 548 324 435 204Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,460.9857177734375,126.17141723632812)"><path d="M51 726L32 700C87 636 187 526 187 266C187 -10 83 -131 32 -194L51 -215C104 -165 273 -23 273 265C273 542 108 675 51 726Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,214.50714111328125,92.17141723632812)"><path d="M146 266C146 526 243 632 301 700L282 726C225 675 60 542 60 266C60 159 85 58 133 -32C168 -99 200 -138 282 -215L301 -194C255 -137 146 -15 146 266Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,220.14996337890625,92.17141723632812)"><path d="M555 243L555 299L51 299L51 243Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,230.4356689453125,92.17141723632812)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,238.92138671875,92.17141723632812)"><path d="M204 123C177 114 159 108 106 93C99 17 74 -48 16 -144L30 -155L71 -136C152 -31 190 32 218 109Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,246.55712890625,92.17141723632812)"><path d="M711 224C711 345 604 366 557 374C637 436 667 482 667 541C667 630 593 689 482 689C414 689 369 670 321 622L292 498L323 498L341 554C352 588 415 622 467 622C532 622 585 569 585 506C585 431 526 368 455 368C447 368 436 369 423 370L408 371L396 318L403 312C441 329 460 334 487 334C570 334 618 281 618 190C618 88 557 21 464 21C418 21 377 36 347 64C323 86 310 109 291 163L264 153C285 92 293 56 299 6C352 -12 396 -20 433 -20C556 -20 711 87 711 224Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,259.27142333984375,92.17141723632812)"><path d="M51 726L32 700C87 636 187 526 187 266C187 -10 83 -131 32 -194L51 -215C104 -165 273 -23 273 265C273 542 108 675 51 726Z" fill="rgb(189,16,224)" fill-opacity="1" stroke="rgb(189,16,224)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,186.91424560546875,54.914276123046875)"><path d="M483 722L471 733C419 707 383 698 311 691L307 670L355 670C379 670 389 663 389 646C389 638 388 629 387 622L359 468C329 477 302 482 277 482C208 482 114 410 68 323C37 265 17 170 17 86C17 21 32 -11 61 -11C88 -11 125 6 159 33C213 77 246 116 313 217L290 126C279 82 274 50 274 24C274 3 283 -9 300 -9C317 -9 341 3 375 28L456 88L446 107L402 76C388 66 372 59 363 59C355 59 349 68 349 82C349 90 350 99 357 128ZM114 59C98 59 89 73 89 98C89 224 135 380 184 418C197 428 216 433 243 433C287 433 316 427 349 410L337 351C300 171 159 59 114 59ZM770 204L741 77C737 60 735 42 735 26C735 4 744 -9 759 -9C782 -9 823 17 905 85L898 106C874 86 845 59 823 59C814 59 808 68 808 82C808 87 808 90 809 93L901 472L891 481L858 463C817 478 800 482 773 482C745 482 725 477 698 464C636 433 603 403 578 354C534 265 503 145 503 67C503 23 518 -11 537 -11C574 -11 654 41 770 204ZM818 414C796 305 777 253 743 201C686 117 625 59 593 59C581 59 575 72 575 99C575 163 603 280 638 360C662 415 685 433 733 433C756 433 774 429 818 414ZM1672 103L1646 86C1617 67 1595 56 1585 56C1578 56 1572 65 1572 76C1572 86 1574 95 1579 116L1654 413C1658 430 1661 446 1661 454C1661 471 1651 482 1635 482C1606 482 1568 464 1507 421C1444 376 1409 341 1360 272L1390 409C1394 429 1397 444 1397 451C1397 470 1387 482 1371 482C1341 482 1295 460 1240 418C1196 384 1176 363 1111 275L1144 408C1148 424 1150 439 1150 451C1150 470 1141 482 1127 482C1106 482 1068 461 994 408L966 388L973 368C1010 391 1045 414 1052 414C1063 414 1070 404 1070 389C1070 338 1029 145 988 2L991 -9L1059 6L1081 108C1105 219 1132 278 1187 338C1229 384 1274 414 1302 414C1309 414 1313 406 1313 393C1313 358 1291 244 1250 69L1234 2L1239 -9L1308 6L1329 113C1345 194 1379 269 1423 318C1478 379 1528 414 1560 414C1568 414 1573 405 1573 389C1573 365 1570 351 1548 264C1508 105 1492 84 1492 31C1492 6 1503 -9 1522 -9C1548 -9 1584 12 1682 85ZM1743 388L1750 368L1782 389C1819 412 1822 414 1829 414C1840 414 1847 404 1847 389C1847 338 1806 145 1765 2L1772 -9C1797 -2 1820 4 1842 8C1861 134 1882 199 1928 268C1982 352 2057 414 2102 414C2113 414 2119 405 2119 390C2119 372 2116 351 2108 319L2056 107C2047 70 2043 47 2043 31C2043 6 2054 -9 2073 -9C2099 -9 2135 12 2233 85L2223 103L2197 86C2168 67 2146 56 2136 56C2129 56 2123 65 2123 76C2123 81 2124 92 2125 96L2191 372C2198 401 2202 429 2202 446C2202 469 2191 482 2171 482C2129 482 2060 444 2001 389C1963 354 1935 320 1883 247L1921 408C1925 426 1927 438 1927 449C1927 470 1919 482 1904 482C1883 482 1844 460 1771 408ZM3006 722L2994 733C2942 707 2906 698 2834 691L2830 670L2878 670C2902 670 2912 663 2912 646C2912 638 2911 629 2910 622L2882 468C2852 477 2825 482 2800 482C2731 482 2637 410 2591 323C2560 265 2540 170 2540 86C2540 21 2555 -11 2584 -11C2611 -11 2648 6 2682 33C2736 77 2769 116 2836 217L2813 126C2802 82 2797 50 2797 24C2797 3 2806 -9 2823 -9C2840 -9 2864 3 2898 28L2979 88L2969 107L2925 76C2911 66 2895 59 2886 59C2878 59 2872 68 2872 82C2872 90 2873 99 2880 128ZM2637 59C2621 59 2612 73 2612 98C2612 224 2658 380 2707 418C2720 428 2739 433 2766 433C2810 433 2839 427 2872 410L2860 351C2823 171 2682 59 2637 59ZM3293 204L3264 77C3260 60 3258 42 3258 26C3258 4 3267 -9 3282 -9C3305 -9 3346 17 3428 85L3421 106C3397 86 3368 59 3346 59C3337 59 3331 68 3331 82C3331 87 3331 90 3332 93L3424 472L3414 481L3381 463C3340 478 3323 482 3296 482C3268 482 3248 477 3221 464C3159 433 3126 403 3101 354C3057 265 3026 145 3026 67C3026 23 3041 -11 3060 -11C3097 -11 3177 41 3293 204ZM3341 414C3319 305 3300 253 3266 201C3209 117 3148 59 3116 59C3104 59 3098 72 3098 99C3098 163 3126 280 3161 360C3185 415 3208 433 3256 433C3279 433 3297 429 3341 414ZM3590 390L3534 107C3533 99 3521 61 3521 31C3521 6 3532 -9 3551 -9C3586 -9 3621 11 3699 74L3730 99L3720 117L3675 86C3646 66 3626 56 3615 56C3606 56 3601 64 3601 76C3601 102 3615 183 3644 328L3657 390L3764 390L3775 440C3737 436 3703 434 3665 434C3681 528 3692 577 3710 631L3699 646C3679 634 3652 622 3621 610L3596 440C3552 419 3526 408 3508 403L3506 390ZM3901 470C3958 543 3996 610 4030 696L4017 709C3981 697 3971 694 3920 679C3914 599 3890 533 3842 456L3857 446ZM4105 148C4100 97 4094 62 4083 16C4121 -2 4155 -11 4190 -11C4239 -11 4285 7 4336 47C4387 87 4408 124 4408 174C4408 228 4376 257 4302 271L4259 279C4199 290 4180 307 4180 348C4180 401 4225 442 4282 442C4323 442 4361 424 4377 397L4377 342L4400 342C4404 377 4408 404 4419 455C4380 474 4352 482 4319 482C4267 482 4206 452 4158 404C4132 377 4121 352 4121 316C4121 260 4152 228 4218 215L4281 203C4325 195 4341 177 4341 138C4341 73 4295 29 4224 29C4189 29 4158 40 4130 64L4130 148ZM5053 330L5076 330C5084 395 5091 432 5100 458C5076 473 5041 482 5004 482C4959 483 4886 463 4829 400C4775 352 4736 241 4736 136C4736 40 4778 -11 4858 -11C4912 -11 4960 9 5015 54L5065 95L5057 115L5042 105C4970 57 4932 40 4897 40C4841 40 4812 80 4812 159C4812 267 4847 371 4896 409C4917 425 4941 433 4972 433C5017 433 5053 414 5053 390ZM5134 152C5134 46 5179 -11 5262 -11C5317 -11 5377 15 5422 57C5484 116 5528 230 5528 331C5528 425 5478 482 5395 482C5291 482 5134 382 5134 152ZM5358 444C5419 444 5452 399 5452 315C5452 219 5421 113 5377 60C5359 39 5333 27 5302 27C5244 27 5210 72 5210 151C5210 264 5249 387 5298 427C5311 438 5334 444 5358 444ZM5577 152C5577 46 5622 -11 5705 -11C5760 -11 5820 15 5865 57C5927 116 5971 230 5971 331C5971 425 5921 482 5838 482C5734 482 5577 382 5577 152ZM5801 444C5862 444 5895 399 5895 315C5895 219 5864 113 5820 60C5802 39 5776 27 5745 27C5687 27 5653 72 5653 151C5653 264 5692 387 5741 427C5754 438 5777 444 5801 444ZM6254 722L6242 733C6190 707 6154 698 6082 691L6078 670L6126 670C6150 670 6160 663 6160 648C6160 645 6160 640 6157 622C6146 567 6073 182 6058 132C6045 82 6039 52 6039 31C6039 6 6050 -9 6069 -9C6095 -9 6131 12 6229 85L6219 103L6193 86C6164 67 6142 56 6132 56C6125 56 6119 66 6119 76C6119 82 6120 89 6123 104Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g></svg> </svg></center>