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Points, Lines, and Planes exercise #18699

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@dddao

Hey, I was working on one of the queue'd up items, the "Points, Lines, and Planes" exercise.

I'm not sure how to make pull requests, but hopefully it goes through. :-)

@dddao

I noticed a few typos for the hints of the first problem. I updated it, but I don't know if it's been changed in the pull request as well.

@Christi

Cute pic, and yes it does update the pull request when you push to a branch you've put in a pull request for.

@dddao dddao 4/08 8:33 PM, typo in the hints section for problem 1, reworded anoth…
…er hint, and changed the solution format for problems 3-5 so that "Yes" choice always comes before "No"
f6a18e2
@beneater beneater commented on the diff
exercises/points_lines_and_planes.html
((64 lines not shown))
+
+ <div class="problems">
+ <div id="Problem_1">
+
+
+
+ <div class="question1">
+ <p>What is another way to name plane <code>\mathcal{<var>R</var>}</code>? <BR></p>
+
+ <div id = "original" class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
@beneater Owner

You could simply do the calculation here, rather than defining all the <var>s above, like this:

line([6 + (0 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + (0 - 6) * sin(TILT) + (0 - 2) * cos(TILT)],
     [6 + (10 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + (10 - 6) * sin(TILT) + (0 - 2) * cos(TILT)],
     {arrows: "->"});
@dddao
dddao added a note

I think the only place where the endPt variables are used are in the graphie sections.
I'm kind of new so I don't know much about general coding practices or efficiency, but is it generally a better idea not to define variables if they're actually constants and can be subbed in the code?

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@beneater beneater commented on the diff
exercises/points_lines_and_planes.html
((113 lines not shown))
+ <div class="set-sol" data-type = "text"> <var>e</var><var>b</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>a</var><var>b</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>e</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>a</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>a</var><var>e</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>a</var><var>c</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>c</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>a</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>a</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>e</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>e</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>c</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>b</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>c</var><var>b</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>b</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>e</var><var>b</var> </div>
@beneater Owner

It might make sense to add a helper function (perhaps to utils/probability.js) that returns all the permutations, then you could do something like:

<div class="set-sol" data-type="text" data-each="permutations([a, b, c, e], 3) AS P"><var>P</var></div>

Where permutations([a, b, c, e], 3) is a function that returns an array of all the permutations of any 3 of [a, b, c, e].

The data-each will insert multiple <div>s, that way you don't have to list them all. Also, 4 nPr 3 = 24 and you've only listed 18, so there are a few missing.

@beneater Owner

Oh wait, it's not 4 nPr 3 since not every permutation is valid -- the points have to be non-colinear, so it's 3 nPr 2 * 3 which is 18, so never mind.

@dddao
dddao added a note

Yeah, it should be 18 or (4 nPr 2)*3.

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dddao added some commits
@dddao dddao 4/09 8PM I forgot about Ben's last hint for Problem 5. But I'm not su…
…re I could connect them in a way that wouldn't be confusing. Any configuration of four points with D included will be not co-planar, but if you connect them, your mind can still perceive them as being a quadrilateral, which can be put on a plane.


Instead, I'll add a second hint about connecting three points at a time to form triangles, and then asking if those triangles are co-planar.
87256ed
@dddao dddao Added a "Now" in there to make it sound like it's a magic trick. e1171db
@cbhl

I'm closing this pull request because it's old and can't be merged automatically. Sorry! Thanks for contributing to Khan Academy!

@cbhl cbhl closed this
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Commits on Apr 6, 2012
  1. @dddao

    first commit

    dddao authored
  2. @dddao
Commits on Apr 9, 2012
  1. @dddao

    4/08 8:33 PM, typo in the hints section for problem 1, reworded anoth…

    dddao authored
    …er hint, and changed the solution format for problems 3-5 so that "Yes" choice always comes before "No"
Commits on Apr 10, 2012
  1. @dddao

    4/09 8PM I forgot about Ben's last hint for Problem 5. But I'm not su…

    dddao authored
    …re I could connect them in a way that wouldn't be confusing. Any configuration of four points with D included will be not co-planar, but if you connect them, your mind can still perceive them as being a quadrilateral, which can be put on a plane.
    
    
    Instead, I'll add a second hint about connecting three points at a time to form triangles, and then asking if those triangles are co-planar.
  2. @dddao
This page is out of date. Refresh to see the latest.
Showing with 575 additions and 0 deletions.
  1. +575 −0 exercises/points_lines_and_planes.html
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575 exercises/points_lines_and_planes.html
@@ -0,0 +1,575 @@
+<!DOCTYPE html>
+<html data-require="math graphie math-format graphie-geometry">
+<head>
+ <meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
+ <title>Points, Lines, and Planes</title>
+ <script src="../khan-exercise.js"></script>
+</head>
+<body>
+ <div class="exercise">
+
+ <div class="vars">
+
+ <!--Much of this variables section was repeated many times, each time when making a new problem...-->
+ <!--Couldn't figure out how to separate this section outside of the problem statement for each new problem since it requires graphie-section. -->
+
+ <var id="TILT">(1/150)*randRange(-100, 100)</var>
+ <var id="TILT2">(1/500)*randRange(-100, 100)</var>
+
+ <!--Theta and r used to vary the position of point E a bit for variety-->
+ <var id="r">(1/150)*randRange(0, 100)</var>
+ <var id="theta">(1/100)*randRange(-314, 314)</var>
+
+ <var id="Plane_edgePt1">[6 + (0 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 0 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)]</var>
+ <var id="Plane_edgePt2">[6 + (10 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 10 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)]</var>
+ <var id="Plane_edgePt3">[6 + (12 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)]</var>
+ <var id="Plane_edgePt4">[6 + (2 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 2 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)]</var>
+
+ <var id="Line_endPt1">[6 + (3 - 6)* cos(TILT+TILT2) - (3 - 2)* sin(TILT+TILT2), 2 + ( 3 - 6 ) * sin(TILT+TILT2) + ( 3 - 2 ) * cos(TILT+TILT2)]</var>
+ <var id="Line_endPt2">[6 + (9 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)]</var>
+
+ <var id="Perp1_endPt1">[6 + (6 - 6)* cos(TILT) - (2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 2 - 2 ) * cos(TILT)]</var>
+ <var id="Perp1_endPt2">[6 + (6 - 6)* cos(TILT) - (6 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 6 - 2 ) * cos(TILT)]</var>
+ <var id="Perp2_endPt1">[6 + (6 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)]</var>
+ <var id="Perp2_endPt2">[6 + (6 - 6)* cos(TILT) - (-3 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -3 - 2 ) * cos(TILT)]</var>
+
+ <var id="Line_behindPlane1_Pt1">[6 + (6 - 6)* cos(TILT) - (0.4 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0.4 - 2 ) * cos(TILT)]</var>
+ <var id="Line_behindPlane1_Pt2">[6 + (6 - 6)* cos(TILT) - (.8 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( .8 - 2 ) * cos(TILT)]</var>
+
+ <var id="Line_behindPlane2_Pt1">[6 + (6 - 6)* cos(TILT) - (1.2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 1.2 - 2 ) * cos(TILT)]</var>
+ <var id="Line_behindPlane2_Pt2">[6 + (6 - 6)* cos(TILT) - (1.6 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 1.6 - 2 ) * cos(TILT)]</var>
+
+ <var id="A">[6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)]</var>
+ <var id="B">[6, 2]</var>
+ <var id="C">[6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)]</var>
+ <var id="D">[6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)]</var>
+
+ <var id="E">[6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)]</var>
+
+ <var id="A_ATTRS">{ r: 0.1, fill: "red", stroke: "none" }</var>
+ <var id="B_ATTRS">{ r: 0.1, fill: "blue", stroke: "none" }</var>
+ <var id="C_ATTRS">{ r: 0.1, fill: "green", stroke: "none" }</var>
+ <var id="D_ATTRS">{ r: 0.1, fill: "orange", stroke: "none" }</var>
+ <var id="E_ATTRS">{ r: 0.1, fill: "purple", stroke: "none" }</var>
+
+ <!--To vary the letters used in the diagram-->
+ <var id="a">randFromArray( [ "A", "F", "K", "U" ] )</var>
+ <var id="b">randFromArray( [ "B", "G", "L", "V" ] )</var>
+ <var id="c">randFromArray( [ "C", "H", "M", "W" ] )</var>
+ <var id="d">randFromArray( [ "D", "I", "N", "X" ] )</var>
+ <var id="e">randFromArray( [ "E", "J", "O", "Y" ] )</var>
+ <var id="R">randFromArray( [ "R", "S", "T", "P" ] )</var>
+
+ </div>
+
+ <div class="problems">
+ <div id="Problem_1">
+
+
+
+ <div class="question1">
+ <p>What is another way to name plane <code>\mathcal{<var>R</var>}</code>? <BR></p>
+
+ <div id = "original" class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
@beneater Owner

You could simply do the calculation here, rather than defining all the <var>s above, like this:

line([6 + (0 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + (0 - 6) * sin(TILT) + (0 - 2) * cos(TILT)],
     [6 + (10 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + (10 - 6) * sin(TILT) + (0 - 2) * cos(TILT)],
     {arrows: "->"});
@dddao
dddao added a note

I think the only place where the endPt variables are used are in the graphie sections.
I'm kind of new so I don't know much about general coding practices or efficiency, but is it generally a better idea not to define variables if they're actually constants and can be subbed in the code?

Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment
+ line( Line_endPt2, Line_endPt1, { arrows: "->" } );
+ line( Perp1_endPt1, Perp1_endPt2, {arrows: "->"});
+ line( Perp2_endPt1, Perp2_endPt2, {arrows: "->"});
+
+ <!--Make it look 3-D-->
+ line( Line_behindPlane1_Pt1, Line_behindPlane1_Pt2);
+ line( Line_behindPlane2_Pt1, Line_behindPlane2_Pt2);
+
+ <!--Draw the points-->
+ circle( A, A_ATTRS );
+ circle( B, B_ATTRS );
+ circle( C, C_ATTRS );
+ circle( D, D_ATTRS );
+ circle( E, E_ATTRS );
+
+ <!--Label the points-->
+ label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], "<var>a</var>", "above right" );
+ label([6.1, 2], "<var>b</var>", "above right");
+ label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], "<var>c</var>", "above right" );
+ label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], "<var>d</var>", "right" );
+ label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], "<var>e</var>", "below" );
+
+ <!--No.w label the ABC line and the plane-->
+ label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right");
+ label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{<var>R</var>}", "right");
+
+ </div>
+ </div>
+
+
+ <div class="solution" data-type="set">
+ <div class="set-sol" data-type = "text"> <var>a</var><var>b</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>a</var><var>e</var><var>b</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>b</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>a</var><var>b</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>e</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>a</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>a</var><var>e</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>a</var><var>c</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>c</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>a</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>a</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>e</var><var>a</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>e</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>b</var><var>c</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>b</var><var>c</var> </div>
+ <div class="set-sol" data-type = "text"> <var>e</var><var>c</var><var>b</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>b</var><var>e</var> </div>
+ <div class="set-sol" data-type = "text"> <var>c</var><var>e</var><var>b</var> </div>
@beneater Owner

It might make sense to add a helper function (perhaps to utils/probability.js) that returns all the permutations, then you could do something like:

<div class="set-sol" data-type="text" data-each="permutations([a, b, c, e], 3) AS P"><var>P</var></div>

Where permutations([a, b, c, e], 3) is a function that returns an array of all the permutations of any 3 of [a, b, c, e].

The data-each will insert multiple <div>s, that way you don't have to list them all. Also, 4 nPr 3 = 24 and you've only listed 18, so there are a few missing.

@beneater Owner

Oh wait, it's not 4 nPr 3 since not every permutation is valid -- the points have to be non-colinear, so it's 3 nPr 2 * 3 which is 18, so never mind.

@dddao
dddao added a note

Yeah, it should be 18 or (4 nPr 2)*3.

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+
+ <div class="input-format">
+ <p>Plane <span class="entry"></span></p>
+ </div>
+ </div>
+
+ <div class="hints">
+ <p>Planes can be named with three <u>non-collinear</u> points. <BR>
+ Non-collinear points are points not on the same line.</p>
+ <p>Find three points in <code>\mathcal{<var>R</var>}</code> not on the same line. List them in any order.</p>
+ <p>For example, we can write <code>\mathcal{<var>R</var>}</code> as plane <var>a</var><var>b</var><var>e</var> or plane <var>a</var><var>c</var><var>e</var> or plane <var>b</var><var>e</var><var>c</var>.</p>
+ </div>
+ </div>
+
+
+ <!--Next problem-->
+ <div id="Problem_2">
+
+ <div class="vars">
+
+ <var id="line_ab"><code>\overleftrightarrow{<var>a</var> <var>b</var>}</code></var>
+ <var id="line_ba"><code>\overleftrightarrow{<var>b</var> <var>a</var>}</code></var>
+ <var id="line_cb"><code>\overleftrightarrow{<var>c</var> <var>b</var>}</code></var>
+ <var id="line_bc"><code>\overleftrightarrow{<var>b</var> <var>c</var>}</code></var>
+ <var id="line_ca"><code>\overleftrightarrow{<var>c</var> <var>a</var>}</code></var>
+ <var id="line_ac"><code>\overleftrightarrow{<var>a</var> <var>c</var>}</code></var>
+
+ <var id="line_a"><code>\overleftrightarrow{<var>a</var>}</code></var>
+ <var id="line_b"><code>\overleftrightarrow{<var>b</var>}</code></var>
+ <var id="line_c"><code>\overleftrightarrow{<var>c</var>}</code></var>
+
+ <var id="seg_ab"><code>\overline{<var>a</var> <var>b</var>}</code></var>
+ <var id="seg_ba"><code>\overline{<var>b</var> <var>a</var>}</code></var>
+ <var id="seg_cb"><code>\overline{<var>c</var> <var>b</var>}</code></var>
+ <var id="seg_bc"><code>\overline{<var>b</var> <var>c</var>}</code></var>
+ <var id="seg_ca"><code>\overline{<var>c</var> <var>a</var>}</code></var>
+ <var id="seg_ac"><code>\overline{<var>a</var> <var>c</var>}</code></var>
+
+ <var id="line_ec"><code>\overleftrightarrow{<var>e</var> <var>c</var>}</code></var>
+ <var id="line_ce"><code>\overleftrightarrow{<var>c</var> <var>e</var>}</code></var>
+ <var id="line_da"><code>\overleftrightarrow{<var>d</var> <var>a</var>}</code></var>
+ <var id="line_db"><code>\overleftrightarrow{<var>d</var> <var>b</var>}</code></var>
+ <var id="line_dc"><code>\overleftrightarrow{<var>d</var> <var>c</var>}</code></var>
+ <var id="line_ad"><code>\overleftrightarrow{<var>a</var> <var>d</var>}</code></var>
+
+
+ <var id="SOLUTIONS">
+ [line_ab, line_ba, line_bc, line_cb, line_ac, line_ca]
+ </var>
+
+ <var id="IDX">randRange( 0, SOLUTIONS.length - 1 )</var>
+ <var id="SOLUTION">SOLUTIONS[ IDX ]</var>
+
+ <var id="fake_SOLUTIONS1">
+ [line_a, line_b, line_c]
+ </var>
+
+ <var id="IDX1">randRange( 0, fake_SOLUTIONS1.length - 1 )</var>
+ <var id="fake_SOLUTION1">fake_SOLUTIONS1[ IDX1 ]</var>
+
+ <var id="fake_SOLUTIONS2">
+ [seg_ab, seg_ba, seg_bc, seg_cb, seg_ac, seg_ca]
+ </var>
+
+ <var id="IDX2">randRange( 0, fake_SOLUTIONS2.length - 1 )</var>
+ <var id="fake_SOLUTION2">fake_SOLUTIONS2[ IDX2 ]</var>
+
+ <var id="fake_SOLUTIONS3">
+ [line_ec, line_ce, line_da, line_db, line_dc, line_ad]
+ </var>
+
+ <var id="IDX3">randRange( 0, fake_SOLUTIONS3.length - 1 )</var>
+ <var id="fake_SOLUTION3">fake_SOLUTIONS3[ IDX3 ]</var>
+
+ </div>
+
+ <div class="question2">
+ <p>What is another way to name line <code>\ell</code>?</p>
+
+ <div class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
+ line( Line_endPt2, Line_endPt1, { arrows: "->" } );
+ line( Perp1_endPt1, Perp1_endPt2, {arrows: "->"});
+ line( Perp2_endPt1, Perp2_endPt2, {arrows: "->"});
+
+ <!--Make it look 3-D-->
+ line( Line_behindPlane1_Pt1, Line_behindPlane1_Pt2);
+ line( Line_behindPlane2_Pt1, Line_behindPlane2_Pt2);
+
+ <!--Draw the points-->
+ circle( A, A_ATTRS );
+ circle( B, B_ATTRS );
+ circle( C, C_ATTRS );
+ circle( D, D_ATTRS );
+ circle( E, E_ATTRS );
+
+ <!--Label the points-->
+ label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], "<var>a</var>", "above right" );
+ label([6.1, 2], "<var>b</var>", "above right");
+ label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], "<var>c</var>", "above right" );
+ label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], "<var>d</var>", "right" );
+ label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], "<var>e</var>", "below" );
+
+ <!--No.w label the ABC line and the plane-->
+ label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right");
+ label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{<var>R</var>}", "right");
+
+ </div>
+ </div>
+
+ <div class "solution">
+ <p class="solution"> <var>SOLUTION</var> </p>
+
+ <ul class="choices">
+ <li><var>fake_SOLUTION1</var></li>
+ <li><var>fake_SOLUTION2</var></li>
+ <li><var>fake_SOLUTION3</var></li>
+ <li><code>\overleftrightarrow{<var>a</var> <var>b</var> <var>c</var>}</code></li>
+
+ </ul>
+ </div>
+
+
+ <div class="hints">
+ <p>You need any two points on the line. </p>
+ <p>Your points must have the <code>\leftrightarrow</code> above. <BR>
+ Lines are named by two points and the <code>\leftrightarrow</code>, and <BR>
+ the order of the letters doesn't matter.</p>
+ <p> <code>\overleftrightarrow{<var>a</var> <var>b</var>}</code>, <code>\overleftrightarrow{<var>b</var> <var>a</var>}</code>, <code>\overleftrightarrow{<var>b</var> <var>c</var>}</code>, <code>\overleftrightarrow{<var>c</var> <var>b</var>}</code>, <code>\overleftrightarrow{<var>a</var> <var>c</var>}</code>, <code>\overleftrightarrow{<var>c</var> <var>a</var>}</code>.</p>
+ </div>
+ </div>
+
+
+ <!--Next problem-->
+ <div id="Problem_3">
+
+ <div class="vars">
+
+ <var id="pts_abc"> <var>a</var>, <var>b</var>, and <var>c</var> </var>
+ <var id="pts_acb"> <var>a</var>, <var>c</var>, and <var>b</var> </var>
+ <var id="pts_bac"> <var>b</var>, <var>a</var>, and <var>c</var> </var>
+ <var id="pts_bca"> <var>b</var>, <var>c</var>, and <var>a</var> </var>
+ <var id="pts_cab"> <var>c</var>, <var>a</var>, and <var>b</var> </var>
+ <var id="pts_cba"> <var>c</var>, <var>b</var>, and <var>a</var> </var>
+ <var id="pts_abd"> <var>a</var>, <var>b</var>, and <var>d</var> </var>
+ <var id="pts_ace"> <var>a</var>, <var>c</var>, and <var>e</var> </var>
+ <var id="pts_bec"> <var>b</var>, <var>e</var>, and <var>c</var> </var>
+ <var id="pts_eca"> <var>e</var>, <var>c</var>, and <var>a</var> </var>
+ <var id="pts_cad"> <var>c</var>, <var>a</var>, and <var>d</var> </var>
+ <var id="pts_cbe"> <var>c</var>, <var>b</var>, and <var>e</var> </var>
+
+ <!--First six points = Yes., second six points = No.-->
+ <var id="points">
+ [pts_abc, pts_acb, pts_bac, pts_bca, pts_cab, pts_cba, pts_abd, pts_ace, pts_bec, pts_eca, pts_cad, pts_cbe]
+ </var>
+
+ <var id="IDX">randRange( 0, points.length - 1 )</var>
+ <var id="oppositeIDX">-IDX + 11 </var>
+
+
+ <var id="sol">
+ [ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.'
+ ]
+ </var>
+
+ <var id="pts">points[ IDX ]</var>
+ <var id="solution">sol[ IDX ]</var>
+
+ </div>
+
+
+ <div class="question3">
+ <p>Are the points <var>pts</var> collinear?</p>
+
+ <div class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
+ line( Line_endPt2, Line_endPt1, { arrows: "->" } );
+ line( Perp1_endPt1, Perp1_endPt2, {arrows: "->"});
+ line( Perp2_endPt1, Perp2_endPt2, {arrows: "->"});
+
+ <!--Make it look 3-D-->
+ line( Line_behindPlane1_Pt1, Line_behindPlane1_Pt2);
+ line( Line_behindPlane2_Pt1, Line_behindPlane2_Pt2);
+
+ <!--Draw the points-->
+ circle( A, A_ATTRS );
+ circle( B, B_ATTRS );
+ circle( C, C_ATTRS );
+ circle( D, D_ATTRS );
+ circle( E, E_ATTRS );
+
+ <!--Label the points-->
+ label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], "<var>a</var>", "above right" );
+ label([6.1, 2], "<var>b</var>", "above right");
+ label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], "<var>c</var>", "above right" );
+ label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], "<var>d</var>", "right" );
+ label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], "<var>e</var>", "below" );
+
+ <!--No.w label the ABC line and the plane-->
+ label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right");
+ label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{<var>R</var>}", "right");
+
+ </div>
+ </div>
+
+ <div class "solution">
+ <p class="solution"> <var>solution</var> </p>
+
+ <ul class="choices" data-category="true">
+ <li><var>sol[0]</var></li>
+ <li><var>sol[11]</var></li>
+ </ul>
+ </div>
+
+
+
+
+ <div class="hints">
+ <p> Collinear means that they lie on the same line.</p>
+ <p> Are the named points all on the same line? </p>
+ </div>
+ </div>
+
+
+ <!--Next problem-->
+ <div id="Problem_4">
+
+ <div class="vars">
+
+ <var id="pts_dc"> <var>d</var> and <var>c</var> </var>
+ <var id="pts_eb"> <var>e</var> and <var>b</var> </var>
+ <var id="pts_ac"> <var>a</var> and <var>c</var> </var>
+ <var id="pts_ca"> <var>c</var> and <var>a</var> </var>
+ <var id="pts_ab"> <var>a</var> and <var>b</var> </var>
+ <var id="pts_ba"> <var>b</var> and <var>a</var> </var>
+ <var id="pts_bd"> <var>b</var> and <var>d</var> </var>
+ <var id="pts_ce"> <var>c</var> and <var>e</var> </var>
+ <var id="pts_ec"> <var>e</var> and <var>c</var> </var>
+ <var id="pts_ca"> <var>c</var> and <var>a</var> </var>
+ <var id="pts_ad"> <var>a</var> and <var>d</var> </var>
+ <var id="pts_be"> <var>b</var> and <var>e</var> </var>
+
+ <var id="points">
+ [pts_dc, pts_eb, pts_ac, pts_ca, pts_ab, pts_ba, pts_bd, pts_ce, pts_ec, pts_ca, pts_ad, pts_be]
+ </var>
+
+ <var id="IDX">randRange( 0, points.length - 1 )</var>
+
+
+ <var id="pts">points[ IDX ]</var>
+ <var id="solution">"Yes."</var>
+ <var id="wrong">"No."</var>
+
+ </div>
+
+
+ <div class="question4">
+ <p>Are the points <var>pts</var> collinear?</p>
+
+ <div class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
+ line( Line_endPt2, Line_endPt1, { arrows: "->" } );
+ line( Perp1_endPt1, Perp1_endPt2, {arrows: "->"});
+ line( Perp2_endPt1, Perp2_endPt2, {arrows: "->"});
+
+ <!--Make it look 3-D-->
+ line( Line_behindPlane1_Pt1, Line_behindPlane1_Pt2);
+ line( Line_behindPlane2_Pt1, Line_behindPlane2_Pt2);
+
+ <!--Draw the points-->
+ circle( A, A_ATTRS );
+ circle( B, B_ATTRS );
+ circle( C, C_ATTRS );
+ circle( D, D_ATTRS );
+ circle( E, E_ATTRS );
+
+ <!--Label the points-->
+ label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], "<var>a</var>", "above right" );
+ label([6.1, 2], "<var>b</var>", "above right");
+ label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], "<var>c</var>", "above right" );
+ label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], "<var>d</var>", "right" );
+ label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], "<var>e</var>", "below" );
+
+ <!--No.w label the ABC line and the plane-->
+ label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right");
+ label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{<var>R</var>}", "right");
+
+ </div>
+ </div>
+
+ <div class "solution">
+ <p class="solution"> <var>solution</var> </p>
+
+ <ul class="choices" data-category="true">
+ <li><var>solution</var></li>
+ <li><var>wrong</var></li>
+ </ul>
+ </div>
+
+
+ <div class="hints">
+ <p> Through any two points, there is exactly one line.</p>
+ <p> Points can be collinear even if the line isn't drawn in the figure. </p>
+ <p> Can you draw a straight line through these two points? <BR>
+ Actually, can you draw a straight line through any two points?</p>
+
+ </div>
+ </div>
+
+
+
+ <!--Next problem-->
+ <div id="Problem_5">
+
+ <div class="vars">
+
+ <!--First six points = Yes., second six points = No.-->
+ <var id="pts_abce"> <var>a</var>, <var>b</var>, <var>c</var>, and <var>e</var> </var>
+ <var id="pts_acd"> <var>a</var>, <var>c</var>, and <var>d</var> </var>
+ <var id="pts_bac"> <var>b</var>, <var>a</var>, and <var>c</var> </var>
+ <var id="pts_bcae"> <var>b</var>, <var>c</var>, <var>a</var>, and <var>e</var> </var>
+ <var id="pts_eda"> <var>e</var>, <var>d</var>, and <var>a</var> </var>
+ <var id="pts_cda"> <var>c</var>, <var>d</var>, and <var>a</var> </var>
+ <var id="pts_abde"> <var>a</var>, <var>b</var>, <var>d</var>, and <var>e</var> </var>
+ <var id="pts_acde"> <var>a</var>, <var>c</var>, <var>d</var>, and <var>e</var> </var>
+ <var id="pts_bedc"> <var>b</var>, <var>e</var>, <var>d</var>, and <var>c</var> </var>
+ <var id="pts_ecda"> <var>e</var>, <var>c</var>, <var>d</var>, and <var>a</var> </var>
+ <var id="pts_cadb"> <var>c</var>, <var>a</var>, <var>d</var>, and <var>b</var> </var>
+ <var id="pts_cbde"> <var>c</var>, <var>b</var>, <var>d</var>, and <var>e</var> </var>
+
+ <var id="points">
+ [pts_abce, pts_acd, pts_bac, pts_bcae, pts_eda, pts_cda, pts_abde, pts_acde, pts_bedc, pts_ecda, pts_cadb, pts_cbde]
+ </var>
+
+ <var id="IDX">randRange( 0, points.length - 1 )</var>
+ <var id="oppositeIDX">-IDX + 11 </var>
+
+
+ <var id="sol">
+ [ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'Yes.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.',
+ 'No.'
+ ]
+ </var>
+
+ <var id="pts">points[ IDX ]</var>
+ <var id="solution">sol[ IDX ]</var>
+
+ </div>
+
+
+ <div class="question5">
+ <p>Are the points <var>pts</var> coplanar?</p>
+
+ <div class="graphie">
+ <!--Draw the plane and the lines-->
+ init({
+ range: [ [-1, 15], [-5, 8] ]
+ });
+ path([ Plane_edgePt1, Plane_edgePt2, Plane_edgePt3, Plane_edgePt4, true ]);
+ line( Line_endPt1, Line_endPt2, { arrows: "->" } );
+ line( Line_endPt2, Line_endPt1, { arrows: "->" } );
+ line( Perp1_endPt1, Perp1_endPt2, {arrows: "->"});
+ line( Perp2_endPt1, Perp2_endPt2, {arrows: "->"});
+
+ <!--Make it look 3-D-->
+ line( Line_behindPlane1_Pt1, Line_behindPlane1_Pt2);
+ line( Line_behindPlane2_Pt1, Line_behindPlane2_Pt2);
+
+ <!--Draw the points-->
+ circle( A, A_ATTRS );
+ circle( B, B_ATTRS );
+ circle( C, C_ATTRS );
+ circle( D, D_ATTRS );
+ circle( E, E_ATTRS );
+
+ <!--Label the points-->
+ label([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], "<var>a</var>", "above right" );
+ label([6.1, 2], "<var>b</var>", "above right");
+ label([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], "<var>c</var>", "above right" );
+ label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], "<var>d</var>", "right" );
+ label([6 + (17/6 + r*cos(theta) - 6)* cos(TILT) - (1.4 + r*sin(theta) - 2)* sin(TILT), 2 + ( 17/6 + r*cos(theta) - 6 ) * sin(TILT) + ( 1.4 + r*sin(theta) - 2 ) * cos(TILT)], "<var>e</var>", "below" );
+
+ <!--No.w label the ABC line and the plane-->
+ label([6 + (9.2 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9.2 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)], "\\ell", "right");
+ label([6 + (12.1 - 6)* cos(TILT) - (4 - 2)* sin(TILT), 2 + ( 12.1 - 6 ) * sin(TILT) + ( 4 - 2 ) * cos(TILT)], "\\mathcal{<var>R</var>}", "right");
+
+ </div>
+ </div>
+
+ <div class "solution">
+ <p class="solution"> <var>solution</var> </p>
+
+ <ul class="choices" data-category="true">
+ <li><var>sol[0]</var></li>
+ <li><var>sol[11]</var></li>
+ </ul>
+ </div>
+
+
+ <div class="hints">
+ <p> Coplanar points are points that lie on the same plane.</p>
+ <p> Pick three of the points and connect them. Now pick another three and connect them. <BR>
+ Do both of the triangles lie on the same plane? </p>
+ <p> Can a flat surface pass through all the points without bending? </p>
+ </div>
+ </div>
+
+ </div>
+ </div>
+</body>
+</html>
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