How Many Solutions?: IM 8.4.8
[url=https://im.kendallhunt.com/MS/teachers/3/4/8/preparation.html]“How Many Solutions?”[/url] from IM Grade 8 by [url=https://www.geogebra.org/m/gus8ffvr]Open Up Resources[/url] and Illustrative Mathematics. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0 license[/url].
Table of Contents
- Lesson 8.4.8
- IM 8.4.8 Lesson: How Many Solutions?
- Practice 8.4.8
- IM 8.4.8 Practice: How Many Solutions?
IM 8.4.8 Lesson: How Many Solutions?
[size=150]Consider the unfinished equation [math]12(x-3)+18=\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore\underscore.[/math][/size][size=150]What is the number of solutions the equation would have with the following expression on the right hand side?[br][br][math]6\left(2x-3\right)[/math][/size]
[math]4(3x-3)[/math]
[math]4(2x-3)[/math]
Here are some cards. With your partner, solve each equation. Then, sort the cards into categories.
Describe the defining characteristics of those categories and be prepared to share your reasoning with the class.
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions).
[size=150][br][math]6x+8=7x+13[/math][/size]
[math]6x+8=2(3x+4)[/math]
[math]6x+8=6x+13[/math]
Looking back at these three equations...
If an equation has one solution, solve to find the value of [math]x[/math] that makes the statement true.[br][br][math]6x+8=7x+13[/math][br][math]6x+8=2(3x+4)[/math][br][math]6x+8=6x+13[/math]
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions).
[math]\frac{1}{4}(12-4x)=3-x[/math]
[math]x-3=3-x[/math]
[math]x-3=3+x[/math]
Looking back at these three equations...
If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]\frac{1}{4}(12-4x)=3-x[/math][br][math]x-3=3-x[/math][br][math]x-3=3+x[/math]
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions).
[math]-5x-3x+2=-8x+2[/math]
[math]-5x-3x-4=-8x+2[/math]
[math]-5x-4x-2=-8x+2[/math]
Looking back at these three equations...
If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]-5x-3x+2=-8x+2[/math][br][math]-5x-3x-4=-8x+2[/math][br][math]-5x-4x-2=-8x+2[/math]
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions).
[math]4(2x-2)+2=4(x-2)[/math]
[math]4x+2(2x-3)=8(x-1)[/math]
[math]4x+2(2x-3)=4(2x-2)+2[/math]
Looking back at these three equations...
If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]4(2x-2)+2=4(x-2)[/math][br][math]4x+2(2x-3)=8(x-1)[/math][br][math]4x+2(2x-3)=4(2x-2)+2[/math]
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of x (and has infinitely many solutions).
[math]x-3(2-3x)=2(5x+3)[/math]
[math]x-3(2+3x)=2(5x-3)[/math]
[math]x-3(2-3x)=2(5x-3)[/math]
Looking back at these three equations...
If an equation has one solution, solve to find the value of x that makes the statement true.[br][math]x-3(2-3x)=2(5x+3)[/math][br][math]x-3(2+3x)=2(5x-3)[/math][br][math]x-3(2-3x)=2(5x-3)[/math]
[size=150][size=100]What do you notice about equations with one solution? [/size][/size]
[size=150][size=100]How is this different from equations with no solutions and equations that are true for every [math]x[/math]?[/size][/size]
Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.
[size=150][size=100]Choose any set of three consecutive numbers. Find their average. What do you notice?[/size][/size]
[size=150][size=100]Find the average of another set of three consecutive numbers. What do you notice?[/size][/size]
[size=150][size=100]Explain why the thing you noticed must always work, or find a counterexample.[/size][/size]
IM 8.4.8 Practice: How Many Solutions?
[size=150][size=100]Lin was looking at the equation [math]2x-32+4(3x-2462)=14x[/math]. She said, “I can tell right away there are no solutions, because on the left side, you will have [math]2x+12x[/math] and a bunch of constants, but you have just [math]14x[/math] on the right side.” Do you agree with Lin? Explain your reasoning.[/size][/size]
[size=150][size=100]Han was looking at the equation [math]6x-4+2(5x+2)=16x[/math]. He said, “I can tell right away there are no solutions, because on the left side, you will have [math]6x+10x[/math] and a bunch of constants, but you have just [math]16x[/math] on the right side.” Do you agree with Han? Explain your reasoning.[/size][/size]
Decide whether each equation is true for all, one, or no values of x.
[size=100][math]6x-4=-4+6x[/math][/size]
[math]4x-6=4x+3[/math]
[math]-2x+4=-3x+4[/math]
Solve each of these equations. Explain or show your reasoning.
[math]3(x-5)=6[/math]
[math]2(x-\frac{2}{3})=0[/math]
[math]4x-5=2-x[/math]
[size=150][size=100]The points [math](-2,0)[/math] and [math](0,-6)[/math]are each on the graph of a linear equation. Is[math](2,6)[/math] also on the graph of this linear equation? Explain your reasoning.[/size][/size]
In the picture triangle [math]A'B'C'[/math] is an image of triangle [math]ABC[/math] after a rotation. The center of rotation is [math]E[/math].[size=100][br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPoAAADRCAYAAADlh6/IAAAgAElEQVR4Ae2dCdxew/XHVfkTkRCJVog0aERoVaSWhJAIsVRCiUhaLRo0rTWkSauWIlRCqaVLCImQRgS1a0tiT2lsFTsl1ugitopUl/l/vsN5PvPOO/feZ7n3Pvd5njOfz/vOfebOds/Mb+bMmTNnVjLqlAJKgaanwEpN/4X6gUoBpYBRoGsniKTAhx9+aJ5//nmzfPnyUpz//e9/pWd9aBwKKNAbp60yrWkIwJMmTTIdOnQw8+fPt2VLHPEzrZBmnioFFOipkrN5MnvqqafM2muvbVZffXUzbdo0o+Bu7LZVoDd2+2VSe0A9atQoM2zYMLP99tubiRMnZlKOZpofBRTo+dG6YUp64IEHzKqrrmoeeughM2LECHPAAQc0TN21omEKKNDDdGnZ0P/+979m8ODBFuAQ4YgjjjADBgwIsu7KzjdON1GgN05b5VLTG2+80ayxxhrmmWeeseWde+65pmfPnuajjz7KpXwtJBsKKNCzoWtD5gqY+/XrZ7baaitz0003GUB/9NFHmzXXXNO89dZbDflNWulPKKBA155QosD06dPNSiutZLp37266detm/7p06WIl748//ngpnj40HgUU6I3XZpnU+J133jEbbrihOfbYYw3P7777rvUffvhhs8oqq5jbb789k3I103wooEDPh86FL+WMM84wnTt3Nm+++WabugJ69tPZS1fXuBRQoDdu26VW85deeskK4E4++eR2ea5YscL06tXLjB8/viR5V2l7OzIVPkCBXvgmyr6Cl112mdltt93M3//+92Bh48aNM8z4OAV5kESFD1SgF76Jsq/gv//978RCXIC7z4kJNUIhKKBAL0QzNHYlFPjFbz8FevHbqLA1VIAXtmnaVUyB3o4kGlAuBf75z38a1vcTJkww8+bNM//5z3/KTarxcqaAAj1ngjdqce7szTNGKYYPH24VbFCy4e+4445TYV1BG1iBXtCGKWq1BPA333xzG5ADdE68cY5dXfEooEAvXps0RI0uv/zydkD/zGc+Y+6///6GqH+rVVKB3motntL3Pvnkk/awi7Dt+OjFv/baaymVoNmkSQEFeprUbLG85syZYzbbbDPTqVMnO7uvvPLK1jIN63ecsPktRpZCfq4CvZDN0jiV+uCDD8xzzz1njVXI7L7ffvtZYV3jfEXz11SB3vxtnMsX3nbbbYY1+s4772yFcvvvv7+CPRfKl1eIAr08OmmsGArAomOCascddzT9+/c3v/jFLyzoXbC7bLz7HJOtvkqRAgr0FInZ6lkxq8O+ozwze/Zs+6xgL0avUKAXox0athbu7MzhmB122MFsvvnmdoZHaw7gA3a06NTVjwIK9PrRvilLlln9yiuvtFJ3MU81cuTIEtjdwaEpiVDAj1KgF7BRGrlK6Lszq/ft27dkOVbAzsyOlF5d/hRQoOdP86YvUWb1WbNm2W9lBhew69ZbfZpfgV4fujdlqcKSM6sjgWdWd29iFbDDxuvMnm8XUKDnS++WKQ2rsQjiZFbnw92ZXQV02XSFZ5991sycOdNMnTrVHiHGTPfHH3+s96NnQ+7WzVVmdSTwMqvLLS/yTqTxsPEqjU+nr2Dv75BDDjEdO3Y066+/vtliiy3MeuutZ9ZZZx3z9NNPK9DTIbPmEqKAaMv5szpxXbALGy8DQSgvDYumwD/+8Q+zzTbbmA022MBce+21dlkELd9//33zyCOP6IweTTp9kwYFRALPvrrM6m6+smZ32XgFu0uh8p4PPvhga3tfbAFAQ6Gj+LpGL4+WGiuBAtKh/GiyVr/iiivsKz+egF3ZeJ9y5f1+7LHHrCzk4osvtuD26Su5KNCFEupnQgGRwHOc1ZXAu4W5bLyu2V3KJD9PmjTJrLXWWpE2+SUHBbpQQv2qKRA1i0iGsq8us7qEu77O7C41yn/m7vpddtklMYECPZFEGqFWCjCrDxo0yBqpiDNKoTN7ZZRG7vGFL3zBHHbYYTZh3ICrQK+Mthq7SgrIWh0d+DgnYHcFdHHxW/kdS6EePXqYY445xpJBgd7KvaEg3y5r9T59+hgubgw56agC9q9//eu6zx4i1Kdh2ADYaqutLLcktIuKrjN6FGU0PHUKMKtjhSZurS6FumBnnz2pI0u6VvO5/BINRPbPfcdAIE6BLpRQP3MKiBUaJPD+vnoIyAL2Vj8IA21C9KHBli1bZnbaaSc7gHKG4LTTTjNHHXWUGThwoHn00UdLbapAL5FCH7KmAJ31rrvusjMQs3pU53XrIdJ4ZeNdqrR9huM599xzza677mr69etn/ZNOOsmgMSdOgS6UUD8XCjCrcxc7a3V/Vo+qgMzsCvYoCiWHK9CTaaQxUqZApbM6xbcq2P/1r3/Za64wqV2pczkmBXql1NP4qVAgNKu7HTNUiFwD1Soz+8svv2zX35/97GfN6quvbkaPHm3ee++9EGkSwxToiSTSCFlQ4J577mkngU8COvWQmX3fffeNNV5RTl5ZfFdaebLEGTNmjJVnIFXnjx2Ls846q1REJd+oQC+RTR/ypACdFOERa/UoHfio+syYMcN2+n322ae0z15Jp4/Kt0jhGIvAQo+AXPw999yzqmoq0KsimyZKgwLM6nTg0L56FHAlnH1jrmlmZo9Tq02jnvXIg+9keSMzuQCdrbNqnAK9GqppmlQoAHs6bNgws+mmm7aZ1QXMFOI++79/+9vfWrAPHz68NLOnUrE6Z8I3P//882aTTTYpzeiw7V/+8pfNX/7yl6pqp0CvimyaqFoK+MC99957bWfGzlklTvJpRrAjYQfk6667rj2CuvXWW5urrroq8ShqHP0U6HHU0XeZUwDAyqwuLLhbqADaDfOfBewjRoxo+JkdkG+00Ub2sApLGqTtl156aTvOxqdB0m8FehKF9H3mFJBZ/Ze//GVVZTEYAPZVVlnFuGAvZ5CoqsCUE0k9seC68cYbW6OOTz75pDnvvPPMaqutZpYsWVJziQr0mkmoGdRKATo6s3qvXr2sQUPyk85fSd6A/f/+7/8s2MXgpKSvJj9JG+enla8L8ocfftgWyck0dibScAr0NKioedRMAZnVuXK5Fnf99ddbAZ07s7v5pQVMN8/QcyXlwK4zk3/+8583AvInnnjCbiFOmzYtlH3FYQr0ikmmCbKgABL43Xffvc2sXm05wsaLNL4S0EWV+bvf/c5yCnAebHthGOPnP/+5eeedd6KSJIZTr1deecX07t3bsusCchKeeeaZljt59dVXg/lU+k0K9CAZNbAeFLj77rutBL7atbpb5xtuuCG49SYAEd9NE/VM3MMPP9x07drVnHLKKebUU081hx56qBWUYa+tHCFiqLzXX3/dbLnlliWQSxyMdMC2M6BImNTN/y3hSb4CPYlC+j4XCkgH9tfq5RQuacWXNCKNl5ldwqvxOf4JuMVRlqjjLlq0SIJjfakfPjO1gJxLFsTxDrYdBZlLLrnEBks6iVONr0CvhmqaJjMKpLFWd4HBzI40XsDuviv3I9566y3TqVMn86tf/apNEuFAYLkryVdm8u7du9ubVNpkaoyZPHlyiW338/V/+2mjfivQoyij4blQwO+4/GatjnVTrhRKw8nM7urGk69fdlRZ999/v1l55ZXN4sWLS1HefvttM3ToUNO/f/8g616K6D24M3logEBWEcW2e1lV9FOBXhG5NHIeFEh7VqfOMrP7YC/ney644ALLSu+99972RBnHZGHlDzroICtMKycP4gDyr3zlK6U1eSidSNuFbQ/FqSZMgV4N1TRNahQIzarurF7t+etQBUUaz9ZbSIAWSkPYN77xDauthlllDpVgww62+/zzz49K0i78tddeSwQ5iWDbUZJBGp+mU6CnSU3NKzUKpDGrhyrjgr2c65+4/pntrxNOOKFNdtddd52d5dl2S3IykyO1D7Hrkp6y0lSSkXzxFeguNfS5MBSQWd3VlkurcoAdqy2iVENZ8ueXgfpphw4dzNVXX21fEQ/HfeRrrLGGmTJlSuxaH5ADXu5Hw4SWpPfL4bfLtsfFC6VNClOgJ1FI32dOgahOzazO8Uz21aPiVFs5d2ZH6BeVP7bokdqjouq6hQsX2hl9zpw5bnCbZ9hv1uRrr722BXmblwFhYFZsO+Uq0H3q6+/CUADwsa+elgSe/OSPjwTsGK8YMmSINY0cAjsKMqimAlrMJy9dutQAfth5jo++++67JXq56f2ZvBQp4iFLtp0iFegRhNfg+lMA4Nx333125qxVBz7qawA7B2EGDx7cxg66xEfCzmCw4YYb2qOjCOE+97nPmVGjRhkEbL6jzlEgdwcCP52vJMP7uPh++qTfCvQkCun7ulKAzr7HHnukOqv7HyQzewjs7J3DprOXzqDzpz/9ybz55pt+FvY3dQX8rMmj2PVgwk+l7Zw9T+NIaqgMBXqIKhpWKAqIBP7iiy/OrF6iG7/zzjsHZ/ZyChaQi+AtlCY0S6Mk89WvfjW1I6mhchXoIapkEBZq4AyKacos057Vo4gUN7NHpZFw2UJjJsfope/i2v+ZZ56xmne+kkxcGj//pN8K9CQKZfA+zQbMoHqFzFJm9azW6vLRsmYvd2anLQXkzOTov1fqpk6dGlSSSbOfKNArbZUq40uj+X6V2bVcMujmr9WFlmkTQ2Z2wM5+eVw5SSB307rPUmfY9m233TZTtp2yFOhC8Rx8aWjxcyiyqYqQffWLLroo8+9KWrPThmy3bbfddlYZhpm8nHb14wjbnpYlmSjCKNCjKFNDOI3pN6hkFxUu79UPU0BompYVmnApn4RKG/lgl3BiMdPvuOOOJY23uPzi3kWx7XFpqnmnQK+GahWmcTsISf3fFWbX0tFlrZ6lBN4lMGBnnx1LMoBy3Lhx5pxzzrGXH3bu3LmqNbnkD9u+zTbbtGHbs+obCnSheo2+NBDX3KI9hS0xCSNrGpX7tNTVRgFoyv1jaWnLubVx28sNv+aaa6zSjlyLhA/477zzTjda6TkqHyK4755++umgtL2UUYoPCvSUiMkVtxxjlBs2uGUDIcv8+fNtCRx1nDBhQkqltXY2oi2Xx6wOMFl/cwjGBTracmICygVvJS3js+3kI3mJX0l+cXEV6HHU+fRdEtEZ2VGL5A4x2Dp0oefOnWvZvAceeMCsWLHCGhbEaqi62ilAe/gS+Npzjc6BCx0F5Byy4Y/ftHO5TvqQ+HlJ26V+CnShRJU+UlOUJDBG4LPrZEnDcjHeeuutZ7h9Q131FBCQ4Oc5q6PxxsEWF+wsHf76179W9DFSfxKJtJ3rllxHHDee+66WZwV6mdSLIj6HHjgz7dv3duPDvqP/rGv0MokdE03ois+szmGTtGzLxRRr1+MDBgywAzbSdji1apzU32fbq8mrkjQK9DKoJY3jR2VUZu2WtK+L0AUlDHXpUoBZHTbaXatLW4mfZokcJWVbDbvrteRP+jyUZNxvV6C71Eh49huXDoZQ5sUXX2yT0o/n/24TWX9UTQHoigS+Z8+euczqUtFa21Ok7b75aMk/C1+BXgNVx44da88oi+2xWjtADVVp2aQcH2XtnKUOfC3tGkoL284EgeyG96E4aTeoAr0Gio4ePdp86UtfsnvkNWSjSWugACBhVmetnqbF2BqqFJtUpO077bRTbiCnQgr02GaJf3nwwQfbGf2jjz6Kj6hva6KAP+v5M6BI4JNkJTVVIqXEsO3IdVy5QkpZx2ajQI8lT/zLX//615ZtLMfkbygnv8OG4mhYMgVkVmetXsusLu0hfnLJn8Tw4/u/3Xxctt0Nz/pZgV4DhbmWp2/fvnaPFXPAmAF64YUXrIQdk0NxLq4zxKXTd2EKyFo9i1ld2kr8cA3ah/rxhW3n+Kv/rn3qdEMU6DXS86mnnrJ3cCFc6dKli1lnnXXs3+zZsxNzzruxEyvUwBEA0V577WUl8JXM6qE2CIVVQpqo9CJtz5ttp+4K9EpaMCIunYxGXLBggXnwwQcNt2+qy5cCgCurtXoUcN1w9znqy+vFtlMfBXpUqwTCy2nMQLLc2bRQHVolDAl8jx49KlqrV9uuUTSV/MQnHpMBR1LrwbZTvgI9qrUiwt3Gc6MQHvXOjcdzufH8dPo7mQJyg8qFF16YHDnHGMK2c+tMPZwCvQqqK1CrIFpGSUJtwQGjLLTlQmWFPisUr55sO3VUoIdaKoMwGt+9vieDIjTLTymwaNEiu+1ZrgTeBSYz70033dTmWmWOGSPVRwYjfxxU4jy6m5Z48+bNM2y7utceC9suSjL1aCgFeg5Ux/wRazOOqmLzjKtz1aVLARdw5Lz//vtXrC3HsVNOIq688soGQyLi2FnhFpU111zT2ojDrDN/3Asnbvny5WbkyJHWLBSKVFtssYV59NFH7Wth2+shbZf6KdCFEhn5nGVef/31S8YK0MvGQAUWRH3nd1b/vf6Op4BLPwZTaF3urE7OWAHi9lOMiLhXI7HuB+TM4OyoyB96FOI4nciMzck03LnnnmsOPPBA+1xvtp1KKNBtU2T37/rrr7cdTowW4HO08o477siu0BbL2QW4++ms1TfYYAMrgZc44rvxeL788svNRhttZGgvDIm4QIeVhxtbtmyZn6zEul9xxRUG2wTirrvuOruvz+96StulPgp0oURG/u9//3sLdDE/BFuIcg3ryKhOJ1VJei/x1P+EAkIvfP5kVr/gggtiSYQ2Izb+MA3FGpzZ2wX69OnTLRcWl8kbb7xhjYtwzfKVV15pDzsBdk6o0eY+ZyF1jcszzXcK9DSpGcjrww8/tKaCZSbHp1O5wppAMg1KiQKyVg8JQgEbxiS4RfW73/2uLZHZe4011mizRp88ebLVdvzhD39ovvOd7xh8BnCEbK579tlnzWGHHWbNiiGUwwnb/tJLL5UG9rxBTj0U6G5LZfBMo7Iep7MgoOHkEqaCd9tttzaSXbfoenQEt/xGf3bpx6wONxW1r3722WebzTbbrKRgc/PNN5uOHTvaGZ18+GOmHzp0qBXwfetb37IXN6yyyip2TY8QLsqRlltSXSUZt25R6bIIV6BnQdWIPG+99VbLxk2aNMmCfcyYMXZGiYiuwVVQIAQkpOFoy/mzOgePEJSKdJzimKkBepLhR5G9wNZHOTEAmaVRjKiy/XAFuk+RDH+zjuvQoYNhzcg6Djb+uOOOy7BEzRoKhNbqLKn69etnBg4caE1z/+Y3vzH8YXt/tdVWsxwAoI9z/fv3t7N6VBzMezPzw7b7LjQg+XHS/K1AT5OaZeSFBdEhQ4bYmKeffroF+5QpU8pIqVFqoQCzOpJz2RJjiwygImlnVpc/Th/C6rPF9rWvfS22yO23394ccsghpbW3Gxkg77DDDnbLLW9Qu/WQZwW6UCInH3Aj1RXW8Mgjj7Qda+bMmbYGRegUOZEi12JkrS6DKnTG1t8HH3xQ+mOW57IGWHds8LP+Jl7IghBsP/KWGTNmtPsO0jz//PP2fYhtr0cbK9DbNVO2AXKU8pZbbrGdiLva2O+Fxavk5o9sa9mcuUNnZvWQshJfDAC5VJHtMNkVQbION4C2G1ZbZ82aZdn7rl272n3zKGEcbLsYgCwCNRXoObcCRhG6detmxo8fXyqZMLmCl5nHdfUY/d3ym+FZaMgsDFsus7p8G+8lDtpvgBr77eIQvLFNx5oeY6DsmDBTM0iHHIMD7Ym0nWfJOxQ3rzAFel6UdsphZqHDiLokr15//XVrloqrflDgUJcNBaA96++4WT1UMmClvaTN3MHBj4+df9h6n22vJ+AV6H4r5fCb002whz6g2Y6BteT6Jtbw9ewYOZAhtyJcOqKRyKyOIou4ONBKnEr8888/326fhqTtleSTZlwFeprULDOvxYsX284WEuRwpxeaWbvuuqsVFrmdlOz932UWqdEcCsCGc2li1KzuRA3SO64NeIe03VWScfOr17MCvQ6UZ23Xu3fvyD1YEQhxmgoVzbiOVYfqN3yRvgQ+zQ9C2o5g1Wfb0yyjmrwU6NVQLYU0RxxxhD0vzZZOyInN+OOPPz70WsNqpACS9HJn9UqKKpq0XequQBdK5OxzsgnNONaMUe4nP/mJjeOuJ6PianhlFJBZHV13cS7n5D6H3kuY65MGTbuise3UUYHutlQOz9KBXn31VWu1BAMF4uSd/GZr5nvf+54FO/u38l58iad++RRwacdaPW5f3c/VTeu/4zfCVdj2elqSCdWLMAV6FGUyDKfDAOLtttvO7snGFYVWFgYN6EC33XZbXFR9VyEF3Fk9CcRJ7ylapO1yS2qF1ck0ugI9U/LGZ/7jH//Y2h4T5YyozsSpq0GDBtm4dM6oePGl6dsQBSqZ1X2689v9Q9qO/ntRlGTc71Wgu9TI+NntKDzfddddli0PXdLoxqVa2J7j3DTXA/v77xlXu6mzZ+BEVuKu1av5YNoEJRlm9SI6BXodW+Wdd96xlksmTpwYWwsBPdZEWVNiwFCvfYolWUUvmdWjtOWYnZcuXWqwBPvQQw8Z1GjxUW6iDXiP4+gxQH/uuecqKjuvyAr0vCgdUQ5HIdGhFtXKiGilYOyLc7pKFGpKL/ShYgrIAOqu1cnkb3/7m70R98QTT7SmozAzdeyxx5qTTz659HfMMccYtkjHjh1rsBO35ZZbWkWZIrLtfJMCveLuEU4gnSb8NjoUE0eccnKNEUbH/uQNhyyYPb75zW+qhZokYpX5nn11zqRjD+7QQw+1B1/QUuT8elTbAmrkK2g4wv6zh15Up0Cvc8s8/vjjtpNg3aQSN23aNJvOVaiRDil+Jfm1elyZ1bEJ9/HHH5fIkURL3nOfGrsiRWXb+RgFeqlJq3tI6gihXN00bJ9h5QRLJZW60047zYL9nHPOKc06bt6V5tfK8aEbszp24MvRgRdakQ4lmQEDBpTW6/KuSL4CPYfWoDPEARCQczw1ZMkkqnqSnyrURFGo8nCMRHKq8Kc//WnZiUW3HWm7tEnZiXOMqEDPkdjSEcSXomHbOTr52GOPSVDZPgdk9t13X8s6sk0neYtfdkYa0VIgTgIfIhEAR8ZSZLadeivQQ62XcxjnlrH1LrbHywGpGweFGthHrhIq5waYnD+voYrDwgyzejn76rQBLLuvJOO2TVE+XoGecUuU0+hsrWGRNMnqaFxVUajp27evvRccFUxcOWXH5dlK71xayck2ttninBiAdNl2N5+4tHm/U6CnRHG2YQ444ABrKwxVSFRW2WO95557SiXQCb7//e/bfddS4KcPKM1galjMEfvvy/mN5VKOXqJQI1Zmy0mncT6hgICUWZ2lVNJaHYAXWUnGbVcFukuNGp7RlmIvFUUKTqSdeuqpdpZm/XbnnXfanDl7DphhC6VTSZFyGSNqsbU4rMyiUIMBQ8wZu84v032nz20pwKyOtpycQ2j71lgJeyNI26XeCnShRI0+FkU6d+7c5tqf999/386whx9+uM395Zdftr/ZO3cdAGQmX2uttQwHXWp13NXNjISFmihLpbWW0Yzp3YFQZvWotbrLtjcCLVoO6KKbHGoct6FD7+PCvv3tb9sZ3M9/4403tmfKSctszUWLUeAbNmyY2Xbbbavej6X+8ocNcjgMzErLd4kf9x2t+i5EG5HA+7M6cWHbi64k47ZlywD9qquustcXY2Z5zz33NO6JsVAju0RKekaYBoCPPvroUlS0q8477zwrTb/33nttONpXl19+eQl4pcifPqD4wr1fGKWotU5kKRZqXOMWhKeRt1/3ZvzNrM5gGVqrw7bz5w/sRaVDSwCdhlp99dUN6qJYauGOa35fc801bTq9AED8chsNm+yw3X369LGScwYSDjlgxN8dUEL5uWWJGiZmpip1kg+++ywKNVzq6DqJ44a18nOIHoSNGjWq3VqdPXNm8yjd9lBe9aZt0wMdEGI+GVbLdayFe/bsWboX231X6fP8+fPtmvioo44yP/rRj+yAwr3YnDCjfGl43/fLQVjHeXNORdXqpCyWCWKhRgYdeVdrGa2QXtbq7qxOX3KVZBqBnk0PdDo3s7dvTB/2uFOnTmbBggU199ezzjrLCtnctTegZYYfMWJECehuQVGdgxNpX/ziF82KFSvc6GU/u/nKMwo1bPmhUAPXoC6aAkIzYsgzs/q6665rdeAJY+vU122XuOJHl1CfN00P9BtvvNHO6ADbdQARIw4IrWp1qKAOHjy4lI009mGHHWYQxmGbPclJGmQJSMwxbJCmc698EoWaNPNv5ryY1dGWQ4aCfgJajBiaaCTX9EBnG4RGuuKKK9q0iwCqVkP7zOIcSMH4gICVghDSoLgC++6Gt6nEpz/c9xiWQAmDGeNnP/tZu71wSe+mkbAkHyspzExc+SRaX9Xkk1ROM74/8MADTZcuXSxnxPoc8DeSa3qgAzjWziiRTJo0yXAxAjPtTjvtZAVonCWuxTGQsF7jfPgbb7xhJeaYG4IFR4J+xx13lJ092zgI8ZD0Mqvjc7ItyvpMNSBFUw+ZBVt5qlBTXtNwhzr9hfaQttlrr73s/enl5VD/WE0PdEjMrHvZZZeZvffe22qMMftyUgwFF1j7WtzNN99sgY4WFRZKWA7wjClntN0qcWy9SWcSHzaRwSRNh1QfLidkoaaawSPNuhUxrz/84Q/t2oWBGC3ERnEtAXQaw+/ASMqZcWu1qIqRAjTdGDjwOdP8yiuv2Pb3ywzVg7hs88FlwFbLjEFHAoywieiwh9LawCr/iUKNa6GmyqyaPtm1115bahcZgPEZABrFtQzQ/QZhywnpaa0KDyEw+2XxW+IxMNBB2IbjxBqDDZ0GKzNId1kHup2JOmKQQtKH8q42DH18ykLIpC6aAlh77dWrV5t24aTgsmXLohMV7E1LAB2QCFDYaqKDs07F+F/WjvUdZ8TRTtt9991LQAbQe+yxh9WeQ7CzfPlyWxUOwHB316abbmpGjx5tXnzxxVIV5RtKARU++On5jYVTwI4ikbowBaATh5ZoL9qFJeCf/9eI7FsAABDWSURBVPznUp8KpypWaEsAfd68eVaYMmTIELvdxUUIt9xyS+otQYdAcIbm1MyZM+2hEtbtAKlDhw5Wko6iDnv3zOw+8NwKoUIb996NW8sz+/X77LOPlTNUKlOopdxGTRslGC3697QE0Dk1hlQciTtrc1/aXGsjIW1HqId0f/PNN7fbY6yvGVA4uYZ5Zlm311pWFum5SILtPLFQk0UZmmd9KdDwQE+a9ZLeR5E/Lh3AuPvuuw1WWNE4Y+uOWRujD5x4uuSSSwy3qriaclHl5B0e9V0o1KDJx1pUFWqiWyWKftEpivGm4YHukjGrRkAYxpoM5RpUWkU6zvbcLrvsYk83Pfjgg4bz567Lqj5uGWk8Sz0ZnNga5OYY/2hmGuVoHvWjQNMAnc4qHTZETved+xwVF3Z/zpw59qQbEnFmbPa0AcEJJ5xgbr311tL9Z0n5hcooSphbd545UrvmmmsGFWr8Ortp/Xf6u1gUaBqg10pWtlA4APODH/zAghltN8ANyDEqgUlmYWnjOnjcu1rrWEv6SuqFQg37+Nxa4uvpV5JPLfUtYlq+3f9+/3cR602dmgroQnTxXaL7YbDZsNtTpkyx+ugIoujcXbt2tdsnHFpAAca/VEHyEd8tQ57j3kmcovuiUDNhwgRb1Vr1DYr+vXH1oz1RrFq8eLFdwnFmIGkPvWh9oKmATmO5BHafmZlYg6IKywEFVFWZsdlPR6DGthd64AjaxLnpJSzOJ36laeLyS/udWze+lX1hjGNcfPHFbeot8bg9lMHPtVAj78RPu45FzI8+ge0CjIvQb9CBwMdMF3oZjUCLpgO621E4msrWFiaW2epiy4uTYWg1YXkFI4quYQg3rfscashyw9x8ivBMvZ999llLD070sSbfZptt7PajXz/iYgSDAZHTfq3qOMNP32FJw1YqZw+wLoPMBm1G9tZD/aFI9GoqoGNJFQOMHFrh9gyUVOikKK3QINOnT7ed3F93FqlB8qgLJ/YOOuigUlF04KjLI1Co4bw9MotG0u0ufVwKD+hgIKD0beXDCdG//vjHP6ZQSrZZBIHuj07+72yrFM49VAfURjlMctFFF9mO2q1bN0t4WCzsmk+dOtVwZBQ11FZzHI/Fdh3GMLnz+7333rM+WnsAltN1sh3IOhxV2CjHehSFGmzSt6KFGmwMcnzYl1PADXJWASOgRXdBoFNpVDCZHZcsWdLmG/hYRjAEErU4H7j+71DexMEkFIYOkYRj8IERFWJzQASJOWqc/sgbyquZw2DJmYG5OQahGoBHz56BkPaEo4ElF7t2WDP129mnD7sS3bt3tzSX3Qc/TjP+pr8Dck4X+g4OkvU6/U5cOf1Y4ubpl4AuFRR/xowZFkQidZVKyUUDrrE8eVerL2WLT35Lly61e9Ycp4TgHNsE3Jtssok59NBD7bYXe97qPqEAh2AQMB577LElkjBzMxtjWcZdtjBgs6UIx+PSvJTQebjtttssC8/94a2kUPPmm29atv3SSy91qPHJI4MfLP0ZZ5zR5l0SLdtEzulHCehSHpVEkojwCi0p1rZuxRHkYFSxXFM6blopI86HxcSc0plnnmm1zigLYKONxuELtNPY5oDjCLlKywvl0chhWNHx73AD3LQnhiZ8F0cveceAAO3pF7RNuQo1fllF/i3f6tcRrhZBXGjJQhh9c+7cuaVkUfmUItTpoR3QqcfkyZNtxxg3bpyVyLprEzTCuKzA318mnfuR7rP7bX44gAW46IejJ46+OMQD4JzF5hICLHnQyULOzy8Up1XCADR26jjeihPasJRhlq/GECZpUftF0iz5IbyjjVCoadTTXOX2CeQ8LHlC/Y9BFQvD3GTrOqGTG1bv53ZAp9Ioj7DOg11hvxCLqeIIYwCIc7Db7D3ywe5HyzMnuRgFWff07t3b7tXCkiM4gkXnCCnbGL6T9H64/E56L/Ga1aczon+PUUmXFhzTZT+8mvP33D6DhRXf+Qo1vHfL9OM36m8MlKBr4DuUqRg8jzzySP9VIX+3AzpHLTnFxEzL0UtOZrkjFhJYV6nE/SoEOhzKZ1bmFJR0OA5IIAVmBGTPllGQGaFHjx5WqMagguaRyzm4+cZ1oLh3bh6t8Ezb0PnYRhQH54XQDY2/JG0uSSM+0nmk8VE0Fgs1rkKNpMWPSufGKfIzHBLmutmuFcc3cXIRWQUCYOwK+K6I390G6GiO0VEweIjjxBagZIsqyUEUQA6A5Y9ZBGspAJp1DlJbJMF0ROyWxxEk7l1SXVr1PaBmJ2LMmDGWpWZAhmtCMozRDZem7nOIXgz03MZKn4hzYqHGv/IpLk2jvGPyoQ/vt99+dqsW81/0Z2QULDPhXBvFtQE6azu2qlijowPOtgEAvemmmxK/h48WwZkA3fVhKYcPH25PfnGzyYUXXmjVUWErmTkYTDCCiNQYtp1OirJGXIeMehcVnvgRFUZIKsd97z5XWExF0RFWov3HMggjGJyZRzmmUhYTk1blGI4UhRr6TTNYqHHbCR0N9DGGDh1q5UVsU8KVVrMEqqgRM4i8knwY0lRU+tjCooPwh6ooYWgA4SRuqB5Iy2VfG4AzEuKjaMHox4xC5yMOrD1KLXALxJO4DCp0GAYMWE3YI9bwsPsQGw0tttTQMYadQlGBbUDUXLEcgxSU/X322tn6YLDA+ENcvUPfkvSt5byvJN9q6heVP+GwlrQZds5wDKD+TTVJZR5zzDFla3yxJEATEc4hJJ2Oq6u8S6qPxMvLj6pPVHhe9aq2HDujIznFQD2jlytFZc0MOGn0chwaagJaQM5an31I1wE8BgWAyP43LDwjJ50TmQA61cxKbK+xh49iB1t8XDhAZyJPDhiw9QeXwMDgcg4MFgwgvENayqWFpOHecfJg0GGwYLY6/fTTLWeBfTfKZitFBgvqBpdCXWUrr5JGriSuSx+e/bT+bz++/Pbj+b/LzZtbXFBI8r87lJ+UjRyHO+MYyJMUasgnLi/JU/30KGCBfsMNN1iwLFy4sF3OXCkEy12uQwFj4sSJ5uyzz7Z20qIatZqGljSovjKLMFggAGSwAKAYXWR2R7jHoIMiA4MFdttGjhxpuQIEU1jyZABgb94dLGSQYgcAWYUMFnReOBy4E058cfpt7NixNm8ZLFijIttgK5C6oG/A7gLbUyisuAOoS0v5Jgnzf0t4yI+L67/jtxvmPvt5s4UqSiBx8fx0TzzxhB2At95667KEVEl58z4pjl+HNH+Hypf6iJ9meVnmtRIzFlpmKES4Tj6E2Y9OHiURd9PIc4hA8s71pQw3jGcJF99/7/6OihMVLmllsIDjgNWH5YfVZW2K3IDjrJxQYo3L7I++M9wApqPYq2awYGkB14BwxucsZLBgicKAwmABd8SRWA6QIDBDkMWgiMwCLgZuBg00Bly4HAYLWG5mWLTXpA38b/N/yzdW45MXWo8ikCUPP3//t1sOAx2DJNxTORp3btpQWf57/V0dBVZCww1hTZTuOrM9HT1k6DCuwZOqE5VWwsV38wmFue/950rj++njfgM69AvYXkF4CLsKDTFmgXCRvWd2F9hiROkHlVTuUUPOgOQWWQisLjsRyCPYxhSrNrIUYbAgHL0GlipsWZKOfV24LDTdOG6LgAjhKVZuMX8lgwV7vSi6cPiCelJfGSzivg0ZiKgVV0NDvh3uCIUauC1oQh1qddXUpdYy3fRSvvjuu6I/t5G6u5WN+5jQu1CYmx/PEkd8/329fifVJ+p9VHi53wE7j+lp9AwAAjsOCM4QjN5+++32qiYUlNDOwggEyisMyoAcmQqgZ5+XY7gIwjiWi4xCBgrAhjCVGZbBAiEoprHQVWewIR/ASL4nnniiHSzQUOQKaK6JAqBsscpgwXKJLbxyvptBjvJFZ4KyZ8+eXS5pNF7KFIgEesrl5J5dOZ1R4oifeyVTKBCBGSwygwVsPqBkrQwLzVqbGZ7z1Mz4J510kt1mY3+c20BZRrCcAPxwFgwWDAouZyGDBcsTdOgZVOBEUBZhJ4UlH4I7BLbkDwfDZZFI/RlkZNDBZwkT0ngUMmC+i4EM4SvcCgo5cCXipJ1YXiF3ScNSreQpZTSr37RAb9YGy+K7ZCcEWQDScwYLAIa5KXQomIlReWXtztl2lgvIGNhX5lw7Zx8YLJi14RzE4IcLcnlmdyXk4BTIgwGEZQ7CTtne9c97855lTGg5Gcpbw5rMOKQ2aLYUiJv9ACq69uwyyFkGATdcAc9wBe5dcm5tkXEgk8C8lziUcVD0QQFITktSB5Yc5W75Sl6t7uuM3uo9oMLv98Hu/3azw1afgJy1OrKGKAfA2blgp8F1cBkoULE7QVn8IZvI4u48t9xme1agN1uLpvw9Ai432zhwu/GQ8KOEhEIS98bHOazwcibCV3umLLYzxaYdwktAHzpMEpd/q79ToLd6D4j4/igw++Hub/fZzZZweSe++55nBHucew+9Z92OKSxxoTjyTv0wBRToYbq0VGgccOLepUUkNAexeyDaeJIvZSNwQ/B28MEHlwaBPOokdWgWX4HeLC3ZYN/hgpW9etby2CzwHUpI6AawRaiuegoo0KunnaZMiQKzZs2y+/eugRPJGoEewjj/9B3v3cFC4qsfpoACPUwXDc2RAoCZPXNfPZeDQmzLcRYAJ8AWP8cqNnxRCvSGb8LG/gBAy4lCtPWQuLNeR0IP+Nk/x486+dfYX55v7RXo+dJbS/MogIINJwA55YfQjePDqNqicXf11Vfb2DKD+76Xlf6MoYACPYY4+ip7CnCiDhNU6OWjbotALkp7jtoI2LOvWXOVoEBvrvZsyK8R8Irvf4SE+74fT39HU0CBHk0bfVNHCgio61iFpipagd5UzdlcH+OD3f/dXF+b7dco0LOlr+auFCgEBRTohWgGrYRSIFsKKNCzpa/mrhQoBAUU6IVoBq2EUiBbCijQs6Wv5q4UKAQFFOiFaAathFIgWwoo0LOlr+auFCgEBRTohWgGrYRSIFsKKNCzpa/m3uIU8JV8+O2H5UEiBXoeVNYympoCAlzx5WM5fTdw4EB7DJejuPxh/46rs/J2CvS8Ka7ltQwFRo8ebS3bYguPW3f549LOau+Qr4VwCvRaqKdplQKfUsCdzXnGWEafPn3s1dpFIJICvQitoHUoNAUALrfPYO2GCyi5EiruDjk+httouZqKu+/EuYOBhOXlK9DzorSW07AUeO+99+zaGvt1WKvljwsquUwi5AA0xjSwXosRjXoCXOqnQBdKqK8UiKAAlm/kHjnXX7BgQbsUAmrW5cQdNGiQ2XXXXc3QoUPNsGHDzAsvvFAX4CvQ2zWVBigF2lLghhtuCAI9ZIdeUnLrTO/evc348eMtq8+lkJMmTTLcMV8Pp0CvB9W1zIaiAPe8YY7anc379etnYOlDbvny5WaDDTYwp5xySuh1XcIU6HUhuxbaKBQQVpxbXtku23rrrc0hhxwSacCS+E8//bRZddVVrbHLonynAr0oLaH1aAgKfPzxx7aeMgCEKj137lx7BfTrr78eep2YPjJRDS8U6DUQT5O2FgXiwC2UIM7xxx9vL41cuHChVY555JFHzKJFi8zbb78t0XL3Fei5k1wLbEQKlANy+S4EcR07djRdunQxa6+9tv3j+b777pMoufsK9NxJrgU2MwUYEJYsWWKeeeaZNn+s8YXtr8f3K9DrQXUtUymQMwUU6DkTXItrTQoI6y9+3lRQoOdNcS2vZSggoBZfPtz/LeFZ+gr0LKmreSsFCkKB/weAqPuteilHbAAAAABJRU5ErkJggg==[/img][br][br][/size]What is the length of side [math]AB[/math]? Explain how you know.
[size=150][size=100]What is the measure of angle [math]D'[/math]? Explain how you know.[/size][/size]