Similar Triangles

<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-similar-triangles" target="_blank">Similar Triangles Exercise</a></p><p><strong><span style="color:#8e44ad;">Similar Triangles</span></strong> are two triangles that share the same three angles. That&#39;s it. For example, these are similar triangles:</p> <center><svg height="148.02142333984375" style=" width:661.8285522460938px; height:148.02142333984375px; 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"><path class="real" d=" M152.6,9.26 L331.6,134 L152.6,134 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M152.6,120.26 L166.34,120.26 L166.34,134" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M400.6,56.65 L511.6,134 L400.6,134 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M400.6,125.48 L409.12,125.48 L409.12,134" 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="146.19285583496094" style="width:660px;height:146.19285583496094px;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(1,0,0,1,155.91424560546875,43.91429138183594)"><path d="M131 331C152 512 241 611 421 665L379 689C283 657 242 637 191 593C88 506 32 384 32 247C32 82 112 -20 241 -20C371 -20 468 83 468 219C468 334 399 409 293 409C216 409 184 370 131 331ZM255 349C331 349 382 283 382 184C382 80 331 13 254 13C169 13 123 86 123 220C123 255 127 274 138 291C160 325 207 349 255 349ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" 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(1,0,0,1,281.91424560546875,130.91427612304688)"><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 224ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" 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(1,0,0,1,403.91424560546875,86.21427612304687)"><path d="M131 331C152 512 241 611 421 665L379 689C283 657 242 637 191 593C88 506 32 384 32 247C32 82 112 -20 241 -20C371 -20 468 83 468 219C468 334 399 409 293 409C216 409 184 370 131 331ZM255 349C331 349 382 283 382 184C382 80 331 13 254 13C169 13 123 86 123 220C123 255 127 274 138 291C160 325 207 349 255 349ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.015300000000000001,0,0,-0.015300000000000001,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,465.914306640625,131.2142761230469)"><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 224ZM762 689C607 689 528 566 528 324C528 207 549 106 584 57C619 8 675 -20 737 -20C888 -20 964 110 964 366C964 585 899 689 762 689ZM744 654C841 654 880 556 880 316C880 103 842 15 750 15C653 15 612 116 612 360C612 571 649 654 744 654ZM1197 689C1115 689 1048 622 1048 539C1048 456 1115 389 1198 389C1279 389 1348 457 1348 537C1348 621 1281 689 1197 689ZM1197 649C1258 649 1308 599 1308 537C1308 479 1258 429 1198 429C1137 429 1088 478 1088 539C1088 600 1137 649 1197 649Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.015300000000000001,0,0,-0.015300000000000001,0,0)"></path></g></g></g></g></svg> </svg></center> <p>But the more interesting bit is what happens when you have two similar triangles. Their sides are proportional to each other. For example, if you have two similar triangles, and one side of the larger triangle is double the corresponding side of the smaller triangle, the other two sides of the larger triangle will also be double their corresponding sides! Wow!</p> <p><strong><span style="color:#e74c3c;">Example</span></strong></p> <p>What is the value of $x$?</p> <p style="text-align: center;">We can use the fact that sides are proportional in similar triangles to solve.</p> <center style="text-align: center;"><svg height="166.67857360839844" style=" width:661.8285522460938px; height:166.67857360839844px; 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"><path class="real" d=" M184.6,9.26 L299.6,134 L120.6,134 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M410.79,51.96 L486.6,134.19 L368.6,134.19 Z" 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="164.85000610351562" style="width:660px;height:164.85000610351562px;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(1,0,0,1,179.91424560546875,34.91429138183594)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414ZM642 689C560 689 493 622 493 539C493 456 560 389 643 389C724 389 793 457 793 537C793 621 726 689 642 689ZM642 649C703 649 753 599 753 537C753 479 703 429 643 429C582 429 533 478 533 539C533 600 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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><g><g><g><g transform="matrix(1,0,0,1,373.91424560546875,93.92855834960938)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" 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> <p>$$\frac{12}{9} = \frac{x}{3}$$</p> <p>$$36x = 9x$$</p> <p>$$x=4$$</p> <p><strong><span style="color:#e74c3c;">Note</span></strong>:&nbsp;It doesn&#39;t really matter how you set the proportion up, as long as you&#39;re consistent. For example, you could set it up the following ways and still get the same answer for $x$:</p> <p>$$\frac{12}{x} = \frac{9}{3}$$</p> <p>$$\frac{9}{12} = \frac{3}{x}$$</p>