It's All on the Line: IM Geo.6.12
[url=https://im.kendallhunt.com/HS/teachers/2/6/12/preparation.html]“It's All on the Line”[/url] from IM Geometry © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Geo.6.12 Lesson
- IM Geo.6.12 Lesson: It's All on the Line
- Geo.6.12 Practice
- IM Geo.6.12 Practice: It's All on the Line
IM Geo.6.12 Lesson: It's All on the Line
Here is line n.
Write an equation for the line that is perpendicular to [math]n[/math] and whose [math]y[/math]-intercept is [math]\left(0,5\right)[/math]. Graph this line above.[br]
Write an equation for the line that is parallel to [math]n[/math] and that passes through the point [math]\left(3,1\right)[/math]. Graph this line above.[br]
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
[table][tr][td]If your teacher gives you the data card:[/td][td]If your teacher gives you the problem card:[br][/td][/tr][tr][td][list=1][*]Silently read the information on your card.[/*][*]Ask your partner, “What specific information do [br]you need?” and wait for your partner to ask for[br]information. Only give information that is on your card. [br](Do not figure out anything for your [br]partner!)[/*][*]Before telling your partner the information, ask, [br]“Why do you need to know (that piece of information)?”[/*][*]Read the problem card, and solve the problem independently.[/*][*]Share the data card, and discuss your reasoning.[/*][/list][/td][td][list=1][*]Silently read your card and think about what[br]information you need to answer the question.[/*][*]Ask your partner for the specific information that [br]you need.[/*][*]Explain to your partner how you are using the[br]information to solve the problem.[/*][*]When you have enough information, share the[br]problem card with your partner, and solve the [br]problem independently.[/*][*]Read the data card, and discuss your reasoning.[/*][/list][/td][/tr][/table]Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
[size=150]Line [math]\ell[/math] is represented by the equation [math]y=\frac{2}{3}x+3[/math]. [/size][br][br]Write an equation of the line perpendicular to [math]\ell[/math], passing through [math]\left(-6,4\right)[/math]. Call this line [math]p[/math].[br]
Write an equation of the line perpendicular to [math]p[/math], passing through [math]\left(3,-2\right)[/math]. Call this line [math]n[/math].[br]
What do you notice about lines [math]\ell[/math] and [math]n[/math]? Does this always happen? Show or explain your answer.[br]
[size=150]Prove that the line [math]Ax+By=C[/math] is always perpendicular to the line that passes through [math]\left(A,B\right)[/math] and the origin.[/size]
IM Geo.6.12 Practice: It's All on the Line
For each equation, is the graph of the equation parallel to the line shown, perpendicular to the line shown, or neither?
[size=150]Main Street is parallel to Park Street. Park Street is parallel to Elm Street. Elm is perpendicular to Willow. How does Willow compare to Main?[/size]
[size=150]The line which is the graph of [math]y=2x-4[/math] is transformed by the rule [math]\left(x,y\right)\longrightarrow\left(-x,-y\right)[/math]. [/size][size=150]What is the slope of the image?[/size]
[size=150]Select [b]all[/b] equations whose graphs are lines perpendicular to the graph of [math]3x+2y=6[/math].[/size]
Match each line with a perpendicular line.
the line through [math](12,4)[/math] and [math](9,19)[/math]
[math]2x-5y=10[/math]
[math]y-4=\frac{2}{3}(x+1)[/math]
[size=150]The graph of [math]y=-4x+2[/math] is translated by the directed line segment [math]AB[/math] shown. What is the slope of the image?[/size][br][br][img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVUAAAD0CAYAAAArO7XRAAAgAElEQVR4Ae2dC/QVRR3HIcQHJK8Me4gghzCQIwiHZyqaiWQqKGagKVCmohmQUREWZIFI8lYRFEEqHnkEEpEIeQiSYjwUgjiaIkSleMwOGeoRnM53au5/79699+7+7u7s7P1/55w9d+/u/HZmvrP3c3/z2Nk6ioEKUAEqQAViU6BObFfihagAFaACVEARqrwJqAAVoAIxKkCoxigmL0UFqAAVqBiqe/bsUVncdu7cqdavXx9py2I5meds3p+sNzfrLcxfRkVQfeedd9SMGTPUunXrMretWbNG/epXvyrYxowZo6ZNm1ZwHHGzWE7kGeXJat6D8r148WI1f/78qioTfkdPPfVU1ZRp2bJlavbs2VVTHtyHc+bMUa+//npZrlYMVcDGxbBv3z61YcOGwKyVOte7d281ePDgQLusHrz//vuzmvXAfL/00ktq06ZNgeeyehC/Izgp1RIAnxUrVlRLcXQ5nnzyydoN1Xnz5qk6dero5r2/Zlu2bKkAT39AdwBssAG81RIIVfdrklB1v45qPVQNIPHpDaVgC9AaqFaTt0qoeu8AN/cJVTfrxZurWg9ViAFAjh07NqcLmldNmjQJbN4bCBuoVpO3SqjmbgFndwhVZ6smlzFCVSnVuHFjNXz48JwoAGwxWMJLRfxmzZqptm3b6v1q8VYJ1dwt4OwOoeps1eQyRqgqpftNTd+p8VK9kDVqwUtFPMT51Kc+pc477zy1Y8cO1bFjx6roWyVUTU27+0mouls3JmeEqlK6mW+gCi8VnmjQCKu339VAFUIirvecETdrn4Sq+zVGqLpfR4SqUro/Fc1946V6+1eLVaEXqsXiZO04oep+jRGq7tcRoaqUwgRkQLWUl+qvSkLVr4h73zlP1b068eeI81T9ioT8Dg8Q/7CuBvSLAqqtWrVSmEoVJhCqYVRKNw6hmq7+YVInVMOoFBDHdagiy4AqJvuHDYRqWKXSi0eopqd92JQJ1bBK+eK5DlUz9xTdAGEDoRpWqfTiEarpaR82ZUI1rFK+eK5DtdjjqL5i5H0lVPPkcPILoepkteRlilDNkyP8F9egivwg4HPIkCG66Y9+1SiBUI2iVjpxCdV0dI+SKqEaRS1PXNegapr76EfFnNSwg1OeIuUm/3uPZX2fU6rcr0FOqXK/jmrtlCqAFZvxWqNWFT3VqIrZj09P1b7mUVOkpxpVsf/Hd81TFRYjz4xQzZPDyS+EqpPVkpcpQjVPjvBfCNXwWqUZk83/NNUPlzab/+F0SjNWrW3+Vyo6PdVKFUzenp5q8hpXmgI9VaGC9FSFwlk2o6dqWXBBcvRUBaJZNqGnKhScnqpQOItm9FQtii1Mip6qUDh6qkLhLJvRU7UsuCA5eqoC0Syb0FMVCk5PVSicRTN6qhbFFiZFT1UoHD1VoXCWzeipWhZckBw9VYFolk3oqQoFp6cqFM6iGT1Vi2ILk6KnKhSOnqpQOMtm9FQtCy5Ijp6qQDTLJvRUhYLTUxUKZ9GMnqpFsYVJ0VMVCkdPVSicZTN6qpYFFyRHT1UgmmUTeqpCwempCoWzaEZP1aLYwqToqQqFo6cqFM6yGT1Vy4ILkqOnKhDNsgk9VaHg9FSFwlk0o6dqUWxhUvRUhcLRUxUKZ9mMnqplwQXJ0VMViGbZpNZ6qlOnThWt+G/qh56qUcLdT3qq7taNyRk9VaNExE/XPFW8PgWvUundu3fEktREJ1RrtHB1j1B1tWZq8kWo1mgRac8lqO7bt081adJEDR48mFD11SKb/z5BHPzK5r+DleLLUq1r/sM7HTt2rN7oqebfDYRqvh4ufiNUXayV/DxZg+r06dPV8uXLU90GDhyoOnTooPPg3S+Vr6VLl6oHH3ywYGvUqJFq06ZNwXHELXU9l89NnDgxs3kP0vWRRx5Rs2fPrqoyTZkyRS1ZsqRqyoQ/iZkzZ1ZNeXAfojzo1igX6pSLUOo8mv9z585VBw8eTG1bvXq1Agiff/55nYeRI0eqnj17ls0Pugt27dpVsJ1yyimqc+fOBccRN81yVpL2tGnTMpv3oHLjbbkrV66sqjLhd7R3796qKdPWrVvV4sWLq6Y8uA/xp2cFqvhHSisA6p06dcob7UcXAJv/+TXC5n++Hi5+Y/PfxVrJz5O15n+aUB0+fLjq169fXskJ1Tw59BdCtVAT144Qqq7VSGF+agVUMX2q1AbARg2cUhVVMfvxOaXKvuZRU+SUqqiK/T++S1OqTBHoqRolaj7pqdZo4eoePVVXa6YmX7XCU60pbs0eoVqjhdkjVI0S7n4Squ7WjclZrYUqnqpCX6s0sPkvVc6eHZv/9rSWpsTmv1A5F5v/wqLkzAjVnBTO7hCqzlZNLmOEak6KaDuEajS90orN5n9ayodPl83/8FqlFbPWNv8rFZyeaqUKJm9PTzV5jStNgZ6qUEF6qkLhLJvRU7UsuCA5eqoC0Syb0FMVCk5PVSicRTN6qhbFFiZFT1UoHD1VoXCWzeipWhZckBw9VYFolk3oqQoFp6cqFM6iGT1Vi2ILk6KnKhSOnqpQOMtm9FQtCy5Ijp6qQDTLJvRUhYLTUxUKZ9GMnqpFsYVJ0VMVCkdPVSicZTN6qpYFFyRX2z1VvLDzggsuyNvGjRunsO6xK4GeqrAm6KkKhbNoRk/VotjCpKJ6qlgDuWXLlvp1SHjMHN8bN26sV6HDouQuBEJVWAuEqlA4i2aEqkWxhUlFhSqW8PSv2YGWMMBayaLzwuwHmhGqgbKUP0ioltco7RiEato1UD79KFCFJwqoYjEkfwBQO3bs6D+cyndCVSg7oSoUzqIZoWpRbGFSUaCK/lRAdceOHXmpwVPFa+f9b/fIi2TxC6EqFJtQFQpn0YxQtSi2MKkoUB08eLBu5nuTAmDx/jk0/10ZrCJUvTUUYZ9QjSBWSlEJ1ZSEj5BsFKhigCrotUhYcB7eqiuBUBXWBKEqFM6iGaFqUWxhUmGhCmgCqPBW0be6bNkyPQMAoG3VqlVBl4AwO7GYEapCGQlVoXAWzQhVi2ILkwoLVTNIBZh6A2BrwOo9nuY+oSpUn1AVCmfRjFC1KLYwqbBQRRMfnmpQMx8j/zjnSiBUhTVBqAqFs2hGqFoUW5hUWKhiZB8eaVDAVCoMVLkSCFVhTRCqQuEsmhGqFsUWJhUWqoBm0JQpM83K/0CAMDuxmBGqQhkJVaFwFs0IVYtiC5MKA1VMlULzHs/84zl/bP3799cDVDiO5n9Qt4AwSxWbEapCCQlVoXAWzQhVi2ILkwoDVcxFBTi9G7xW9LO68ry/t/iEqleNCPuEagSxUopKqKYkfIRkw0A1wuWciEqoCquBUBUKZ9GMULUotjApQlUoHPo7sA6kCwH9M3E8zkaoulCbpfNAqJbWx4WzaNrPmjXLhazEloda4akC6kOGDNGLLqBjGxsWYAha7SassoRqWKXSi0eopqd9uZSffvppdckll+jfovlNTpgwoZxZJs7XCqjCM8Xjbd7VbcxUDO+xKDVGqEZRK524hGo6updLFU9FGZD6PwcMGFDO3PnztQKqxWoBk4kxgigJhKpENbs2hKpdvcOmdvLJJxeFKiC7evXqsJdyMp41qD766KPq3//+tzPbwYMHdcUuWrSoZJ7efvttdejQoYKtefPmqkePHgXHEdelckbJy8yZMzOb96BybtmyRa1du7aqyoTfEe7doPJm4djmzZtLAhVQHTp0aGbLhzqAJ44BuHKhogdr0ac5ffp0hRsi7e2uu+7Sr2OAl9q5c+ey+Zk/f76aMmVKwdaoUSPVunXrguOIm3YZpemPHz8+s3kPKjNeZOjKfReUP8mxe+65R82dOzez9XTrrbeWhWrbtm0zWz7UKe45K1B1YfTfLLyAf0M8nVHJxGE2/8v9D6d/ns3/9OsA4xkbNmzIPQXVvn37slD9wQ9+kH7GK8iBtea/C1A1OsFzNgNV/qXETJxyn4RqOYXSP0+o2qsDLzwx0wZOi38QKuz3nTt32st4AinVSqgaHTEjAN6rJBCqEtXs2hCqdvWGg2JeeVIMoHXr1tWwxQIpQXFGjx5tN9MJpFaroWreGy7RlVCVqGbXhlC1q7c3NUxVvPjii1X9+vVz8KxXr57ex/zww4cPq5/+9KfqjDPO0HPGr732Wj2o6L1GVvdrBVRRwWjyewOO4QEAdANIAqEqUc2uDaFqV2+T2vLly1X37t01QE888cQcVOGZ+h+44WOqRrWInwBamn2qpv8U/TxmQwVXsgYjoRrxJkghOqFqV3QMSPXt2zcHUQw4XXrppbnvfqAid4SqsI7ShiqyjY50jPajYvGJ75UEQrUS9ezYEqr2dB44cGAOnjfffLP+feF3b/pNizkwhKqwjlyAqjDrRc0I1aLSOHOCUE22Kvbv36+GDRuWA+c111yjtm/fnkvUPI6KwatigVAtpkyZ44RqGYEcOY3J8tUUCNVkahODTBilNyP5ffr0UevWrStIDDAtBVQYEKoFsoU7QKiG0yntWIRq2jVQPn2MTeD3lFaYNGmSatq0qfZOu3btqh5//PGiWQmzrgahWlS+0icI1dL6uHKWUHWlJornIy2ozpkzRz+WjT7Sz33uc/pR2eK5VKHBT6iWUrHEOUK1hDgOnSJUHaqMIlmxDdXHHntMr5EBmJ5yyinq3nvvLZIz2WFCVaab/tdKc0qVMNslzThQVVIeJ06yT1VeDVhE+qKLLtLNfEzav/POO9W7774rv2ARS0K1iDDlDtNTLaeQG+fpqbpRD6VykbSnunXrVnX11VfnRvRvu+02vdRgqTxVco5QFapHqAqFs2xGqFoWXJBcUlB99dVX1Y033piD6XXXXad27dolyGE0E0I1ml652IRqTgqndwhVp6tHZy5uqGIR9lGjRuVgiiegNm3aZE0IQlUoNaEqFM6yGaFqWXBBcnFB9aOPPlJYlNy82qRXr17qiSeeEOSoMhNCVagfoSoUzrIZoWpZcEFycUD1gQceUC1atNDeabt27dSCBQsEOYnHhFAV6kioCoWzbEaoWhZckFwlUF24cKE6++yzNUw//elP69d+CLIQqwmhKpSTUBUKZ9mMULUsuCA5CVRXrVqlzj//fA1TLMWHdUw/+OADQerxmxCqQk0JVaFwls0IVcuCC5KLAtXnnntO9evXLzcINWLECPXGG28IUk3OhFAVakuoCoWzbEaoWhZckFwYqO7du1cvZGKW3cOiJjjmYiBUhbVCqAqFs2xGqFoWXJBcKai++eabCt6ogSm8VHirLgdCVVg7hKpQOMtmhKplwQXJBUEV/aPoJzWvLkH/KfpRsxAIVWEtEapC4SybEaqWBRck54fq9OnTFUby4Z1iZB8j/FkKhKqwtghVoXCWzQhVy4ILkjNQxdxSzDEFTDHnFHNPsxgIVWGtEapC4SybEaqWBRckd8cdd6hu3bppmOJpKDwVdezYMcGV3DAhVIX1QKgKhbNsRqhaFjxCcnge3/tmUjyvj+f2sx4IVWENEqpC4SybEaqWBQ+RHFaKwopRZkT/wgsvVDt27AhhmY0ohKqwnghVoXCWzQhVy4KXSO7gwYMKa5kamGKNU6x1avpUS5hm6hShKqwuQlUonGUzQtWy4AHJYXV9rLKP1fYBVKy+v2bNmlxMQjUnhbM7Tz75pH5LbLkM1ikXodR5QrWUOu6cI1TTrQu8/wnvgQJMu3TpovB+KH8gVP2KuPe9VkEVfVHz589X06ZNq7hfiu+ocu9m9ucoK++omjt3rmrTpo2GaevWrRXeXFosEKrFlHHneK2AKjzl/v37q8aNG6vevXvrDftDhgwR1wShKpbOmqHrUF26dKnq2rWrhmnTpk3VpEmTympDqJaVKPUItQKqUHns2LF57yKH14pmlnQklVBN/d4tmwFXobpu3TrVp08fff/VrVtXjR49Wh0+fLhseRCBUA0lU6qRag1Ug1SGtzp16tSgU2WPEaplJUo9gmtQxR/4Nddco2GKP/Rhw4ap/fv3R9KJUI0kVyqRrUEVgyB//OMfndnWr1+vb+4HH3ywZJ6wys+KFSsKNjTX2rdvX3AccV0qZ5S8TJ48ObN5Dyrn8uXL1aJFi1IvE979dNVVV+VgCi9Vmi/8jjZu3Jh6mYL0lhxbvXq1euSRR6qmPNBg3rx5dkb/cTNs2bLFmQ2v48VCFOXytHnzZoUfp39r0qSJhqr/OL6Xu6ar5zH67GreJPlCnyXgJbGNw2bt2rXqhhtuyMG0R48eatasWRXlB78jOARx5M+Fa2A1LQzUuZCXuPKAPwnMvy0XqmpKlfFSpf2pEIvN/3K3TPrn02z+T5w4UQ+MopnfvXt3/acchyJs/sehYrLXsNb8x83gQgBI4WXCRa8kEKqVqGfHNg2oojupVatW2js988wzK77P/EoRqn5F3Pteq6AaF1BRjYSqezezP0c2obpkyRLVqVMnDdPmzZuLB0D9ZfB/J1T9irj3vdZANU6gohoJVfduZn+ObEAVAy1Y5ATN/Pr166uf/OQn6siRI/6sxPadUI1NysQuVCugaoCKyf4bNmwo2CTqEqoS1ezaJAnVF154QV155ZW5Qajbb79d/f3vf0+8gIRq4hJXnECtgCom/sOTKLZJVCRUJarZtUkCqq+88or6xje+kbuXrr/+erV7925rBSNUrUktTqhWQFWsTglDQrWEOI6cihOqb731lsKq++aP+bLLLlOYbmc7EKq2FY+eHqEaXTNtQagKhbNoFgdUjx49qn72s5+phg0baqCee+65Cj+atAKhmpby4dMlVMNrlReTUM2Tw8kvlUL1vvvuU5/97Gc1TM866yz93H3aBSVU066B8ukTquU1CoxBqAbK4tRBKVQBrg4dOmiYAqozZ850plyEqjNVUTQjhGpRaUqfIFRL6+PC2ahQXblypULzHv2mDRo00M3+Dz/80IWi5PJAqOakcHaHUBVWDaEqFM6iWVioYsDp8ssv1zAFUL/73e+qQ4cOWcxp+KQI1fBapRWTUBUqT6gKhbNoVg6qe/bsUZgSZUb0MVXq5ZdftpjD6EkRqtE1s21BqAoVJ1SFwlk0KwZVTNL/zne+k4MpJvFjhaIsBELV/VoiVIV1RKgKhbNo5ofqe++9px8jPf744zVQL7jgAoXHTLMUCFX3a4tQFdYRoSoUzqKZF6p4w8Opp56qYYqFTxYvXmwxJ/ElRajGp2VSVyJUhcoSqkLhLJoBqj/60Y8UluBDv2nLli0VlubLciBU3a89QlVYR4SqUDhLZngDw9lnn61hineR3X333ZZSTjYZQjVZfeO4OqEqVJFQFQqXsNkzzzyj+vbtq2EK7/Taa6/Ne4tuwsknfnlCNXGJK06AUBVKSKgKhUvIDE39QYMG5WB60003qaeeekpt2rQpoRTTuSyhmo7uUVIlVKOo5YlLqHrESHH3wIED+lXPZq4pXgG9fft2nSPvQFWKWYw1aUI1VjkTuRihKpSVUBUKF5PZ4cOH9SBU3bp1tXd68cUXq3Xr1uVdnVDNk8PJL3jrKF7rXk2BUBXWJqEqFC4Gs0mTJqlmzZppmHbt2lU9/vjjgVclVANlceogoSqsjnfeeceJZdOE2Q80I1QDZUn04EMPPaRat26tYdqmTRv18MMPl0yPUC0pjxMnCVVhNRCqQuEsm91///2WUwyX3GOPPaa6dOmiYfqJT3xC3XvvvaEMCdVQMqUaiVAVyk+oCoWzbOYaVJ9++mn1pS99ScO0Xr16asyYMerdd98NrQqhGlqq1CISqkLpCVWhcJbNXIHqtm3b1NVXX61hilH92267Tf31r3+NrAahGlky6waEqlByQlUonGWztKH66quvqm9961s5mGLi/q5du8QqEKpi6awZEqpCqQlVoXCWzdKC6j//+U81atSoHEwvvfRStXHjxopLT6hWLGHiFyBUhRITqkLhLJvZhupHH32kJkyYoE4++WQN1J49e6onnngitlITqrFJmdiFCFWhtISqUDjLZjah+sADD6jTTz9dw7Rdu3bq0Ucfjb20hGrsksZ+QUJVKCmhKhTOspkNqC5atCi3ehTm+k6fPj2xUhKqiUkb24UJVaGULkF1x44dCiu+47OSwMn/0dRbtWqV6t27t/ZMTzjhBDVu3Dj1/vvvR7tIxNiEakTBUohOqApFdwWq8+bN0wsVY5rO+vXrhaX5nxmhGk6+559/XvXr1y83CDVixAj1xhtvhDOuMBahWqGAFswJVaHILkB17NixCosVw0MlVIMrMs7m/969e9WQIUNyMB08eLD685//HJxwQkcJ1YSEjfGyhKpQTBegCpgiHwiEanBFxgHVN998U40cOTIH0yuuuEI999xzwQkmfJRQTVjgGC5PqApFBMwwIIG1IF3YAFU88hgmLwsWLFDTpk0r2Bo1aqQX9wg6F+a6LsbB9CZpvubPn68GDBig6tevr4GK90Jh7qn0enHYzZo1S82cOTPVPMRRDu81fvGLXyh0Y3mPZXkfi+RMmTKlasqDugDr8GdRLtQpF6HUeUAVN8K//vUvJzZAFWsehsnPoUOHFN4T798++clPqu7duxccR7ww13UxzowZM0R5v+eee3JvKj3rrLP06lEulA/9uWvWrBGVyYX8B+UBvyMszB10LovHdu/erZduzGLei+V56dKldqAKgrsS2PwPromozX948e3bt9eeaYsWLVRU++BcxHeUzf/4tEzqSmz+C5V1oU/Vm3VC1atGzX5YKGKl9l69emmYfvzjH1fjx49Xx44dq7mQI3uEqiMVUSIbhGoJcUqdIlRLqePOuXJQxUv0vvKVr+QGob73ve+pt99+250C+HJCqPoEcfAroSqsFEJVKJxls2JQ/dOf/qSuu+66HExvvPFG9Ze//MVy7qInR6hG18y2BaEqVJxQFQpn2cwP1YMHD6pvf/vbOZhidH/r1q2WcyVPjlCVa2fLklAVKu0aVPEgwL59+4Sl+Z9ZNT9R9Z///Efdeeed6rjjjtNA/eIXv6hH0SsSLAVjQjUF0SMmSahGFMxEdw2qJl+VfFYrVCdPnqwwXQyDeZ07d1a/+c1vKpEpVVtCNVX5QyVOqIaSqTASoVqoiWtH5s6dq5o3b65hesYZZ6g5c+a4lsXI+SFUI0tm3YBQFUpOqAqFs2CGicrdunXTMG3QoIHCRP5qCYSq+zVJqArriFAVCpegGVbp6tOnj4YpmvqjR49WaPpXUyBU3a9NQlVYR4SqULgEzLCwzNe+9rUcTG+55Ra1f/9+nZJ/9D+B5K1eklC1KrcoMUJVJJvSq0O59JiqsBh5ZlkbqMLNe/PNN+dgOnDgQPXiiy/mlYlQzZPDyS/4HcFJqZZAqAprkp6qULgYzLDoww9/+MMcTC+55BK1YcOGwCsTqoGyOHWQUHWqOgIzg8Wa8GdRLlS8ShU91XISx39+4sSJqkmTJhqoWFFr2bJlJRMhVEvK48RJQtWJaiiZCUK1pDzFT7rc/J89e7bCtCgMQLVt21Yvu1i8JDVnCNUaLVzdI1RdrZmafBGqNVpE2pNCFd4iXjyIdTHjDkuWLFHnnHOOhikm8GPx3yiBUI2iVjpxCdV0dI+SKqEaRS1PXClUW7ZsqaE3fPhwz9Uq2/3973+v8CgpPFOsvP/jH/9YHTlyJPJFCdXIklk3IFStSx45QUI1smT/M5BA1bx8sGPHjvp1zcKkc2YvvPCCuuqqqzRMAdTbb79d/e1vf8udj7pDqEZVzH58QtW+5lFTJFSjKvb/+FGhihkQGDQCWPFm0U6dOglTVuqVV15R3/zmN3Mw/frXv67wWopKA6FaqYLJ2xOqyWtcaQqEqlDBqFA1Xirgin14llHDW2+9pe64444cTC+77DL17LPPRr1M0fiEalFpnDlBqDpTFUUzQqgWlab0iShQxTKDgChgijB16lT9Pewk7qNHj6qf//znqmHDhtruC1/4gn5xYekcRj9LqEbXzLYFoWpb8ejpEarRNdMWUaCK5n7jxo1zT8LguXtAFp/lwn333adOO+00HR9vKv3lL39ZzkR8nlAVS2fNkFC1JrU4IUJVKF1YqOJZe6+XiuQMVEtNxv/1r3+tOnTooG0/85nP6PfXC7Ma2oxQDS1VahEJ1dSkD50woRpaqvyIYaHau3dvDUaA1b+Z7gDvlVeuXKnOO+88Hfekk05Sd911l/rwww+9URLbJ1QTkza2CxOqsUmZ2IUIVaG0YaBqPFLAE/veDYD1zlX9wx/+oC6//PIceEeOHKkOHTokzJ3MjFCV6WbTilC1qbYsLUJVppsKA1XMR8Vk/6Bg5qru2bNH3XDDDTmYDh06VL388stBJokfI1QTl7jiBAjViiVM/AKEqlDiclDFY6jwRos9jtqzZ8/cu6AQr3///mrLli3C3MRjRqjGo2OSVyFUk1Q3nmsTqkIdy0EVHir6U/3hvffe01Or6tWrp6GLdQB+97vf+aOl8p1QTUX2SIkSqpHkSiUyoSqUvRRUMaoPoKIP1RumTZumTj31VA3TFi1aqHbt2nlPp75PqKZeBWUzQKiWlSj1CISqsApKQdV/yfnz56vPf/7zGqbwYGfNmuWP4sR3QtWJaiiZCUK1pDxOnLQC1YMHD6oZM2Y4UeC4MhEGqr/97W9Vjx49NEwbNWqk7r777riST+Q6kyZNSuS6aV109erVCsshVlPA7wi/p2oJ27dvVw899FC1FEeX4+GHH1YoV7kQ/UF1zxVxE+DRzGoKzZo1K7ooyjPPPKO+/OUva5hiEOr73/9+7mkqlzVwHfpRtatGqOJ3RKhGvRPsxidUI+ptFkMBLM1mvPCdO3eqQYMG5Y7fdNNN6rXXXouYQnrRCdX0tA+bMqEaVqlk4+FJyWKBUC2mTMBx73xSA1Tz2a1btxxMv/rVr4Zy/wOSSPUQoZqq/KESJ1RDyZR4JEAVc80xXuIPhKpfkSLfzTP8BqJBnxdeeKFau3ZtkSu4f5hQdb+OCFV36t+pdTcAAAj1SURBVMjMRW/VqlUeXK1B9fzzz9ePZeLRTBe3W2+9VfeRYvHooA0DU0Eg9R7r27evk2ULq3evXr0ynX9/OfFABfq2/cez/B2/IyxQnuUyePOO7rKLLroos+XBtEjDAANXa1Dt0qVLLnGTCX7W9MtSC2rBeyB798Dxxx+fxzVMmRw1alSo7r+KR//Hjx+vRy0xcuniduDAAbV169ai27hx4/LEC/oBYAqVi2ULm6fRo0dnOv/+ci5cuFDPCfYfz/J33Ie4T7NcBm/eMUMDU/m8x7Kyj3U78BAPWACYmkfSrXmq6AvKcjh27Jhq0KBBUbBicn/WA/tU3a9B9qm6U0f9+vXLg6nJGaFqlFCqYC6p/3Un8HyCPFT0t27evNlzpWzuEqru1xuh6kYd4VF045n6c0So/h+m6DyHSAApmlh486kBKBY9MYD9xz/+oc/j3HHHHacmT56s3n//fb+umfxOqLpfbYSq+3VU66Fq5pvhE+D0LoSCF/ZhLhoACrB6w8c+9jH93invsazvE6ru1yCh6n4dWYPq3LlznVPDABUwxYapVP73RuE4XtoHsAKyJlQjVKdPn26KVxWfGzduVKtWraqKsphC4HeEgZxqCXhGPmvrM4AJGzZsyOMB6sMcx8s6t23bVraKKhr9R2JYXcelAECiiQ+wIqD5H7T+qTkHqHqX8qtGqHKVKpfu0OC8cJWqYF1sH8VoP96S7A3mrcl4bP3111/3ngrcrzqoYuTOiALAwhs1gPUrYJ6c8HqxhKpfJfe+v/TSS2rTpk3uZayCHBGqFYgXoym6YeBowWFEMN/heFlZ+s81TxXwhCAGosifESdIdwhFTzVIGbePEapu1w9yB49uxYoV7mfUl0PwAo6Yeakn+GBmA9RKqKKpDxHCBrMyVbX3qbL5H/aOSC8ePdX0tPenDI7g0VR0I5pWL+JYhyoAhcykGdB3Wqz/NChfEAyzALyh2pr/+MNAGc2/rbesWdtHywLP/eOBjHPOOUeNGDGiZEvE9fKhboYMGaJnoDRs2FCde+65eQt4uJ7/Uvkzniq649LmQql8Bp0zLV4/S6xBdfbs2fqmgMvsz0RQhpM8FiUPcPPxT+SHDRap9oM2yTwneW1ACGU88cQTdXMmybSSvjbKgvpFHxemtkyYMEHXk39KXNL5iPP6uM/wx46yjRkzRqFFYcoYZzppXAtQveKKK3R50uZC1PKjXtDihbfqDdagetJJJ+VujLTFCxLCK4p3Hz/OoPyGeZ2K9zqu7hsIYRCuTZs2mYcqdDb946ZP1XgU3u4bV+ujXL5M8x9eXdB9Wc7etfOLFi3Sj39nrTz4k8MfG34/3v5U6GsNqniTKAJ+vGnfDEgfQpT7keE8pk6YAS3vDVktUAWATPmqBaqmngxUUUbUtymnOZ/Fz2qCKurltNNO0943ugXT5kLY+8E70g8bdF14vVVrUMXNgACypy2eGXhCv1uxgApHPov9EKsFqt7yVytU0XWDP8dqCPgdofmPPwnvFL8slg0wGjp0qB79zwpUoTm0B1hNMN4qPhFqJVQBTLjuEAcDAPjuDfgRAqilPFlC1auYm/vwVPFEFfqLvT8CN3NbOle4J/HEH+7Z008/PfNABZzQJ4n3umFKVRagCk4UG1DDcZQBIVaoHj16VC1fvjy3mZWbkBmXPFUUHB6o6WjGjYqBDGw45h+UCrrdCdUgVdw69uyzz+p+Yu90F7dyGD43+IOHJ4SBqmHDhmV6oAplwR8dfoNm9D8LUA1bW7FC9ciRI/omRjMSW58+fXQ+XISqEQgVi5sVWynP1MQ3n4SqUcLNT9xzmFKF16lUUzB9qqYZinJmLaAViK43PD+P5/6xkA9ajPDEg56pz1r5YoVqscK7DNVieS53nFAtp1B653G/4QeKqTrV+pgqnAC0sIr1+aenfvmUke9Sm2lGl7+SmzEIVWG9EKpC4RI2M0BFk9+M/iecZKKX93uixlMFeDAukPXA5r+wBumpCoWzbFYNo/8YMDB95F27dtVPVJn+8qw9sYPqh0eK6TpoHmPx9AEDBugnqlDGrI/+o3yEqvBH7oUq+i3DDAQJk7JmVo2e6vXXX69/xNZETCAh3Fvw4rDdcsstesqO+Z7V+w7wRBnwh3HllVfqp8Si9P8nIHNslzRQxZ9HVuvHL4b15r8/A1n9Xo1Q5YIq7t+Npvnvfk7D5dBANVzsbMQiVIX1RKgKhbNoVg19qn65CFW/Iu59J1SFdUKoCoWzaEaoWhRbmBQ9VaFw3j5V4SWcMyNUnauSggwRqgWSOHeAUBVWCaEqFM6yGftULQsuSI7Nf4Folk3Y/BcKTk9VKJxFM3qqFsUWJkVPVSgcPVWhcJbN6KlaFlyQHD1VgWiWTeipCgWnpyoUzqIZPVWLYguToqcqFI6eqlA4y2b0VC0LLkiOnqpANMsm9FSFgtNTFQpn0YyeqkWxhUnRUxUK55Knirzg1S6VPuZHqApvBotmhKpFsYVJEapC4VyBKpZJM6un41njSgKhWol6dmwJVTs6V5IKoSpUzwWoYrEG86pprPBDqBZWJvtUCzVx7Qj7VF2rkcL81Jo+Vaz0Y0BKqBbeCDhCqAbr4tJRQtWl2gjOizWozpgxQ0MNYEt7A1TxIrgw+VizZo1auHBhwda0aVN15plnFhxH3DDXdTEO+ppdzJc0T4sXL1YLFiyoqjLhd4SXGUo1cc0O77SbPXt21ZQH+s6ZM0evExuM3JqjdWp2ZXu7d+9WrmyAKroDwuTnxRdfjFzhYa7LOO7cD6wL1kXc90AYSlYM1TCJ2IoTR/PfVl6ZDhWgAtWpQGagOmjQIHXCCSfktgMHDhTUCKFaIAkPUAEqYFmBzED1tddeU9u2bcttH3zwQYFUhGqBJDxABaiAZQUyA9UwuhCqYVRiHCpABZJUgFBNUl1emwpQgVqnQFVBtXfv3gpPVzFQASpABdJSoKqgmpaITJcKUAEqYBQgVI0S/KQCVIAKxKAAoRqDiLwEFaACVMAoQKgaJfhJBagAFYhBAUI1BhF5CSpABaiAUYBQNUrwkwpQASoQgwL/BVJfTvQmUDmHAAAAAElFTkSuQmCC[/img]
[size=150]Select [b]all[/b] points on the line with a slope of [math]-\frac{1}{2}[/math] that go through the point [math]\left(4,-1\right)[/math].[/size]
[size=150]One way to define a circle is that it is the set of all points that are the same distance from a given center. How does the equation [math]\left(x-h\right)^2+\left(y-k\right)^2=r^2[/math] relate to this definition? Draw a diagram if it helps you explain.[/size]