Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

295 lines (257 sloc) 14.78 kb
<!DOCTYPE html>
<html data-require="math graphie word-problems subhints">
<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">
<var id="TILT">randRange(-100, 100) / 150</var>
<var id="TILT2">randRange(-100, 100) / 500</var>
<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>
<p class="question"></p>
<div class="problem">
<div class="graphie">
init({
range: [ [-1, 15], [-5, 8] ],
scale: [ 30, 30 ]
});
// plane
path([
[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)],
[6 + (12 - 6) * cos(TILT) - (4 - 2) * sin(TILT), 2 + (12 - 6) * sin(TILT) + (4 - 2) * cos(TILT)],
[6 + (2 - 6) * cos(TILT) - (4 - 2) * sin(TILT), 2 + (2 - 6) * sin(TILT) + (4 - 2) * cos(TILT)],
true
]);
style({ arrows: "-&gt;" }, function() {
// line in the plane
line([6 + (3 - 6)* cos(TILT+TILT2) - (3 - 2)* sin(TILT+TILT2), 2 + ( 3 - 6 ) * sin(TILT+TILT2) + ( 3 - 2 ) * cos(TILT+TILT2)],
[6 + (9 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)]);
// line in the plane (opposite direction hack to get double-headed arrow)
line([6 + (9 - 6)* cos(TILT+TILT2) - (1 - 2)* sin(TILT+TILT2), 2 + ( 9 - 6 ) * sin(TILT+TILT2) + ( 1 - 2 ) * cos(TILT+TILT2)],
[6 + (3 - 6)* cos(TILT+TILT2) - (3 - 2)* sin(TILT+TILT2), 2 + ( 3 - 6 ) * sin(TILT+TILT2) + ( 3 - 2 ) * cos(TILT+TILT2)]);
// perpendicular line (top)
line([6 + (6 - 6)* cos(TILT) - (2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 2 - 2 ) * cos(TILT)],
[6 + (6 - 6)* cos(TILT) - (6 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 6 - 2 ) * cos(TILT)]);
// Perpendicular line (bottom)
line([6 + (6 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)],
[6 + (6 - 6)* cos(TILT) - (-3 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -3 - 2 ) * cos(TILT)]);
});
// dashed line behind the plane
style({ strokeDasharray: "- " }, function() {
line([6 + (6 - 6)* cos(TILT) - (2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 2 - 2 ) * cos(TILT)],
[6 + (6 - 6)* cos(TILT) - (0 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( 0 - 2 ) * cos(TILT)]);
});
var r = randRange(0, 100) / 150;
var theta = randRange(-314, 314) / 100;
// points
style({ r: 0.15, stroke: "none" }, function() {
circle([6 + (4 - 6)* cos(TILT+TILT2) - (8/3 - 2)* sin(TILT+TILT2), 2 + ( 4 - 6 ) * sin(TILT+TILT2) + ( 8/3 - 2 ) * cos(TILT+TILT2)], { fill: PINK });
circle([6, 2], { fill: BLUE });
circle([6 + (8 - 6)* cos(TILT+TILT2) - (4/3 - 2)* sin(TILT+TILT2), 2 + ( 8 - 6 ) * sin(TILT+TILT2) + ( 4/3 - 2 ) * cos(TILT+TILT2)], { fill: GREEN });
circle([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], { fill: ORANGE });
circle([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)], { fill: "purple" });
});
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)], A, "above right" );
label([6.1, 2], B, "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)], C, "above right" );
label([6 + (6 - 6)* cos(TILT) - (-2 - 2)* sin(TILT), 2 + ( 6 - 6 ) * sin(TILT) + ( -2 - 2 ) * cos(TILT)], D, "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)], E, "below" );
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{" + R + "}", "right");
</div>
</div>
<div class="problems">
<div id="name-plane">
<p class="question">
What is another way to name plane <code>\mathcal{<var>R</var>}</code>?
</p>
<div class="solution" data-type="set">
<div class="set-sol" data-type="text"><var>A + B + E</var></div>
<div class="set-sol" data-type="text"><var>A + C + E</var></div>
<div class="set-sol" data-type="text"><var>A + E + B</var></div>
<div class="set-sol" data-type="text"><var>A + E + C</var></div>
<div class="set-sol" data-type="text"><var>B + A + E</var></div>
<div class="set-sol" data-type="text"><var>B + C + E</var></div>
<div class="set-sol" data-type="text"><var>B + E + A</var></div>
<div class="set-sol" data-type="text"><var>B + E + C</var></div>
<div class="set-sol" data-type="text"><var>C + A + E</var></div>
<div class="set-sol" data-type="text"><var>C + B + E</var></div>
<div class="set-sol" data-type="text"><var>C + E + A</var></div>
<div class="set-sol" data-type="text"><var>C + E + B</var></div>
<div class="set-sol" data-type="text"><var>E + A + B</var></div>
<div class="set-sol" data-type="text"><var>E + A + C</var></div>
<div class="set-sol" data-type="text"><var>E + B + A</var></div>
<div class="set-sol" data-type="text"><var>E + B + C</var></div>
<div class="set-sol" data-type="text"><var>E + C + A</var></div>
<div class="set-sol" data-type="text"><var>E + C + B</var></div>
<div class="input-format">
<p>Plane <span class="entry"></span></p>
</div>
</div>
<div class="hints">
<div>
<p>Planes can be named with three <a href="#" class="show-definition" data-definition="noncollinear">noncollinear</a> points.</p>
<div class="definition" id="noncollinear">
Noncollinear points are points that are not on the same line.
</div>
</div>
<p>
Find any three points in the plane <code>\mathcal{<var>R</var>}</code> that are
not on the same line and list them in any order.
</p>
<p>
For example, we can write <code>\mathcal{<var>R</var>}</code> as plane
<code><var>A + B + E</var></code>, plane <code><var>A + C + E</var></code>,
or plane <code><var>B + E + C</var></code>.
</p>
</div>
</div>
<div id="line-name">
<div class="vars">
<var id="SOLUTION">randFromArray([
"\\overleftrightarrow{" + A + B + "}",
"\\overleftrightarrow{" + A + C + "}",
"\\overleftrightarrow{" + B + A + "}",
"\\overleftrightarrow{" + B + C + "}",
"\\overleftrightarrow{" + C + A + "}",
"\\overleftrightarrow{" + C + B + "}"
])</var>
<var id="INCORRECT_1">randFromArray([
"\\overleftrightarrow{" + A + "}",
"\\overleftrightarrow{" + B + "}",
"\\overleftrightarrow{" + C + "}"
])</var>
<var id="INCORRECT_2">randFromArray([
"\\overline{" + A + B + "}",
"\\overline{" + B + A + "}",
"\\overline{" + C + B + "}",
"\\overline{" + B + C + "}",
"\\overline{" + C + A + "}",
"\\overline{" + A + C + "}"
])</var>
<var id="INCORRECT_3">randFromArray([
"\\overleftrightarrow{" + E + C + "}",
"\\overleftrightarrow{" + C + E + "}",
"\\overleftrightarrow{" + D + A + "}",
"\\overleftrightarrow{" + D + B + "}",
"\\overleftrightarrow{" + D + C + "}",
"\\overleftrightarrow{" + A + D + "}"
])</var>
</div>
<p class="question">What is another way to name line <code>\ell</code>?</p>
<div class="solution"><code><var>SOLUTION</var></code></div>
<ul class="choices">
<li><code><var>INCORRECT_1</var></code></li>
<li><code><var>INCORRECT_2</var></code></li>
<li><code><var>INCORRECT_3</var></code></li>
<li><code>\overleftrightarrow{<var>A + B + C</var>}</code></li>
</ul>
<div class="hints">
<p>Lines are named using any two points on the line. The order doesn't matter.</p>
<p>The points must have the <code>\leftrightarrow</code> above because we're
naming a line, not a ray or a segment.</p>
<p>Another way to name line <code>\ell</code> is <code><var>SOLUTION</var></code>.</p>
</div>
</div>
<div id="collinear-3">
<div class="vars">
<var id="POINTS,SOLUTION">randFromArray([
[[A, B, C], "Yes"],
[[A, B, D], "No"],
[[A, C, B], "Yes"],
[[A, C, E], "No"],
[[B, A, C], "Yes"],
[[B, C, A], "Yes"],
[[B, E, C], "No"],
[[C, A, B], "Yes"],
[[C, A, D], "No"],
[[C, B, A], "Yes"],
[[C, B, E], "No"],
[[E, C, A], "No"]
])</var>
</div>
<p class="question">Are the points <var>toSentence(POINTS)</var> collinear?</p>
<div class="solution"><var>SOLUTION</var></div>
<ul class="choices" data-category="true">
<li>Yes</li>
<li>No</li>
</ul>
<div class="hints">
<p>Collinear means that they lie on the same line.</p>
<p>Can you draw a straight line through points <var>toSentence(POINTS)</var>?</p>
<p data-if="SOLUTION === 'Yes'">Yes, points <var>toSentence(POINTS)</var> are collinear.</p>
<p data-else>No, points <var>toSentence(POINTS)</var> are not collinear.</p>
</div>
</div>
<div id="collinear-2">
<div class="vars">
<var id="POINTS">shuffle([A, B, C, D, E], 2)</var>
</div>
<p class="question">Are the points <var>toSentence(POINTS)</var> collinear?</p>
<div class="solution">Yes</div>
<ul class="choices" data-category="true">
<li>Yes</li>
<li>No</li>
</ul>
<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 points <var>toSentence(POINTS)</var>?
Actually, can you draw a straight line through any two points?</p>
<p>Yes, points <var>toSentence(POINTS)</var> are collinear.</p>
</div>
</div>
<div id="coplanar">
<div class="vars">
<var id="POINTS,SOLUTION,COPLANAR">randFromArray([
[[A, B, C, E], "Yes", []],
[[A, B, D, E], "No", [A, B, E]],
[[A, C, D], "Yes", []],
[[A, C, D, E], "No", [A, C, E]],
[[B, A, C], "Yes", []],
[[B, C, A, E], "Yes", []],
[[E, C, D, A], "No", [E, C, A]],
[[B, E, D, C], "No", [B, E, C]],
[[C, B, D, E], "No", [C, B, E]],
[[C, D, A], "Yes", []],
[[E, D, A], "Yes", []]
])</var>
</div>
<p class="question">Are the points <var>toSentence(POINTS)</var> coplanar?</p>
<p class="solution"><var>SOLUTION</var></p>
<ul class="choices" data-category="true">
<li>Yes</li>
<li>No</li>
</ul>
<div class="hints">
<p>Coplanar points are points that all lie on the same plane.</p>
<p>Can a flat surface pass through all the points without bending?</p>
<p data-if="SOLUTION === 'No'">
No, any flat surface that includes three of the points won't include the fourth.
For example, points <var>toSentence(COPLANAR)</var> are in plane
<code>\mathcal{<var>R</var>}</code>, but point <var>D</var> is not.
</p>
<p data-else-if="POINTS.length === 4">
Yes, points <var>toSentence(POINTS)</var> all lie within a single
flat surface. In this case, plane <code>\mathcal{<var>R</var>}</code>.
</p>
<p data-else>
Yes, there is always at least one flat surface that passes through
any three points.
</p>
</div>
</div>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.