This repository has been archived by the owner on May 11, 2021. It is now read-only.
/
areas_of_trapezoids_rhombi_and_kites.html
257 lines (221 loc) · 13 KB
/
areas_of_trapezoids_rhombi_and_kites.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
<!DOCTYPE html>
<html data-require="math graphie word-problems subhints graphie-geometry">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Areas of trapezoids, rhombi, and kites</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="UNIT,UNIT_TEXT">randFromArray([
["in", "inch"],
["ft", "foot"],
["m", "meter"],
["cm", "centimeter"],
["", "unit"]
])</var>
</div>
<div class="problems">
<div id="trapezoid">
<div class="vars">
<var id="B1">randRange(2, 8)</var>
<var id="B2">randRange(2, 8)</var>
<var id="H">randRange(1, 6)</var>
<var id="SH">randRangeNonZero(-2, 2)</var>
<var id="K">1/2 * (B1 + B2) * H</var>
</div>
<div class="question">What is the area of this figure?</div>
<div class="graphie" id="figure">
init({
range: [[-4, max(B1, B2) + 4], [-1, H + 1]],
scale: [30, 30]
});
var v = [[0, 0], [B1, 0], [B2 + SH, H], [SH, H], [0, 0]];
style({ stroke: BLUE, fill: "#eee"},
function() {
path(v);
label([B1/2, 0], B1 + "\\text{ " + UNIT + "}", "below");
label([B2/2 + SH, H], B2 + "\\text{ " + UNIT + "}", "above");
path([[B1, 0], [B1, H]], H, { strokeDasharray: "." });
label([B1, H/2], H + "\\text{ " + UNIT + "}", "right");
parallel([[0, 0], [B1, 0]], 1);
parallel([[SH, H], [B2 + SH, H]], 1);
});
rightAngleBox([[0, 0], [B1, 0]], [[B1, 0], [B1, H]], { stroke: GRAY, opacity: 0.5 });
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>K</var></span>
square <var>plural(UNIT_TEXT)</var>
</div>
<div class="hints">
<p>This figure is a quadrilateral with a pair of parallel sides (the top and bottom sides), so it's a trapezoid.</p>
<div>
<p>
area of a trapezoid <code>= \dfrac12 \cdot (b_1 + b_2) \cdot h</code>
[<a href="#" class="show-subhint" data-subhint="area-trapezoid">Show me why</a>]
</p>
<div class="subhint" id="area-trapezoid">
<p>Let's draw a line between the opposite ends of the two bases.</p>
<div class="graphie" data-update="figure">
var showSubHint = function() {
graph.subhint.show();
$("a[data-subhint='area-trapezoid']")
.unbind("click", showSubHint)
.click(hideSubHint);
};
var hideSubHint = function() {
graph.subhint.hide();
$("a[data-subhint='area-trapezoid']")
.unbind("click", hideSubHint)
.click(showSubHint);
};
graph.subhint = raphael.set().push(
path([[0, 0], [B1, 0], [B2 + SH, H]], { stroke: BLUE, fill: ORANGE, opacity: 0.5 }),
path([[SH, H], [B2 + SH, H], [0, 0]], { stroke: BLUE, fill: RED, opacity: 0.5 })
);
hideSubHint();
</div>
<p>Notice that the line divides the trapezoid into two triangles: one triangle with base <code>b_1 = <var>B1</var></code>, and another triangle with base <code>b_2 = <var>B2</var></code>. Both triangles have height <code>h = <var>H</var></code>.</p>
<p>The area of the trapezoid is equal to the sum of the areas of the two triangles.</p>
<p><code>A = \dfrac12 \cdot b_1 \cdot h + \dfrac12 \cdot b_2 \cdot h</code></p>
<p>Factor out <code>\dfrac12 \cdot h</code> to get the formula for the area of a trapezoid:</p>
<p><code>A = \dfrac12 \cdot h \cdot (b_1 + b_2) = \dfrac12 \cdot (b_1 + b_2) \cdot h</code></p>
</div>
</div>
<div>
<p>Now use this formula to calculate the trapezoid's area.</p>
<p><code>b_1 = <var>B1</var></code></p>
<p><code>b_2 = <var>B2</var></code></p>
<p><code>h = <var>H</var></code></p>
<p><code>A = \dfrac12 \cdot (<var>B1</var> + <var>B2</var>) \cdot <var>H</var> = <var>K</var></code></p>
</div>
</div>
</div>
<div id="kite">
<div class="vars" data-ensure="D1 > SH">
<var id="D1">randRange(1, 7) * 2</var>
<var id="D2">randRange(1, 7) * 2</var>
<var id="ORIENT">randFromArray(["v", "h"])</var>
<var id="SH">rand(3) !== 0 ? randRange(1, 5) : D1/2</var>
<var id="K">1/2 * D1 * D2</var>
<var id="RHOMBUS">SH === D1/2</var>
</div>
<div class="question">What is the area of this figure?</div>
<div class="graphie" id="figure">
var range, v;
var drawCongruencies, drawD1, drawD2;
if (ORIENT === "h") {
range = [[-1, D1 + 2], [-D2/2 - 1, D2/2 + 1]];
v = [[0, 0], [SH, D2/2], [D1, 0], [SH, -D2/2], [0, 0]];
drawCongruencies = function(style) {
congruent([[0, 0], [SH, D2/2]], 1, style);
congruent([[0, 0], [SH, -D2/2]], 1, style);
congruent([[SH, D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style);
congruent([[SH, -D2/2], [D1, 0]], RHOMBUS ? 1 : 2, style);
};
drawD1 = function(style) {
return {
label: label([D1/2, 0], D1 + "\\text{ " + UNIT + "}", style),
path: path([[0, 0], [D1, 0]], style)
};
};
drawD2 = function(style) {
return {
label: label([D1, 0], D2 + "\\text{ " + UNIT + "}", "right", style),
path: path([[D1, -D2/2], [D1, D2/2]], style)
};
};
} else {
range = [[-D2/2 - 1, D2/2 + 1], [-1, D1 + 2]];
v = [[0, 0], [D2/2, SH], [0, D1], [-D2/2, SH], [0, 0]];
drawCongruencies = function(style) {
congruent([[0, 0], [D2/2, SH]], 1);
congruent([[0, 0], [-D2/2, SH]], 1);
congruent([[D2/2, SH], [0, D1]], RHOMBUS ? 1 : 2);
congruent([[0, D1], [-D2/2, SH]], RHOMBUS ? 1 : 2);
};
drawD1 = function(style) {
return {
label: label([0, D1/2], D1 + "\\text{ " + UNIT + "}", style),
path: path([[0, 0], [0, D1]], style)
};
};
drawD2 = function(style) {
return {
label: label([0, D1], D2 + "\\text{ " + UNIT + "}", "above", style),
path: path([[-D2/2, D1], [D2/2, D1]], style)
};
};
}
init({ range: range, scale: 20 });
path(v, { stroke: BLUE, fill: "#eee" });
drawCongruencies({ stroke: BLUE });
style({ stroke: BLUE, strokeDasharray: "." }, function() {
graph.d1 = drawD1();
graph.d2 = drawD2();
});
rightAngleBox(graph.d1.path.graphiePath, graph.d2.path.graphiePath, { stroke: GRAY, opacity: 0.5 });
</div>
<div class="solution" data-type="multiple">
<span class="sol"><var>K</var></span>
square <var>plural(UNIT_TEXT)</var>
</div>
<div class="hints">
<div>
<p>This figure is a quadrilateral with perpendicular diagonals and two pairs of congruent, adjacent sides, so it is a kite.</p>
<p data-if="SH === D1/2">In fact, because this shape's sides are all congruent, it is also a rhombus.</p>
</div>
<div>
<p>
area of a kite <code>= \dfrac12 \cdot d_1 \cdot d_2</code>
[<a href="#" class="show-subhint" data-subhint="area-kite">Show me why</a>]
</p>
<div class="subhint" id="area-kite">
<p>The <span data-if="ORIENT === 'h'">horizontal</span><span data-else>vertical</span> diagonal in the center splits the kite into two congruent triangles.</p>
<div class="graphie" data-update="figure">
var showSubHint = function() {
graph.subhint.show();
$("a[data-subhint='area-kite']")
.unbind("click", showSubHint)
.click(hideSubHint);
};
var hideSubHint = function() {
graph.subhint.hide();
$("a[data-subhint='area-kite']")
.unbind("click", hideSubHint)
.click(showSubHint);
};
if (ORIENT === "h") {
graph.subhint = raphael.set().push(
path([[0, 0], [SH, D2/2], [D1, 0], [0, 0]], { fill: ORANGE, opacity: 0.5 }),
path([[0, 0], [SH, -D2/2], [D1, 0], [0, 0]], { fill: GREEN, opacity: 0.5 })
);
} else {
graph.subhint = raphael.set().push(
path([[0, 0], [D2/2, SH], [0, D1], [0, 0]], { fill: ORANGE, opacity: 0.5 }),
path([[0, 0], [-D2/2, SH], [0, D1], [0, 0]], { fill: GREEN, opacity: 0.5 })
);
}
hideSubHint();
</div>
<p>Let <code>d_1 = <var>D1</var></code>, the diagonal that bisects the kite. Then let <code>d_2 = <var>D2</var></code>.</p>
<p>Notice that <code>d_1</code> is the base of both triangles, and <code>d_2</code> is the combined height of the two triangles, so <code>\dfrac{d_2}{2}</code> is the height of each triangle.</p>
<p>So the area of each triangle is:</p>
<p><code>A_T = \dfrac12 \cdot b \cdot h = \dfrac12 \cdot d_1 \cdot \dfrac{d_2}{2} = \dfrac14 \cdot d_1 \cdot d_2</code></p>
<p>The area of both triangles combined, <code>2A_T</code>, is the total area of the kite:</p>
<p><code>2A_T = 2(\dfrac14 \cdot d_1 \cdot d_2) = \dfrac12 \cdot d_1 \cdot d_2 = A</code></p>
</div>
</div>
<div>
<p>Now use this formula to calculate the kite's area.</p>
<p><code>d_1 = <var>D1</var></code></p>
<p><code>d_2 = <var>D2</var></code></p>
<p><code>A = \dfrac12 \cdot <var>D1</var> \cdot <var>D2</var> = <var>K</var></code></p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>