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

322 lines (269 sloc) 15.088 kb
<!DOCTYPE html>
<html data-require="math graphie graphie-helpers graphie-geometry math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Solving similar triangles 2</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="side-a">
<div class="vars" data-ensure="ANGLE < 45">
<var id="HEIGHT_A">randRange(1, 10)</var>
<var id="HEIGHT_B">HEIGHT_A + randRange(1, 20)</var>
<var id="SIDE_B">randRange(1, 10)</var>
<var id="SIDE_A">SIDE_B * HEIGHT_A / (HEIGHT_B - HEIGHT_A)</var>
<var id="LABELS">["x", HEIGHT_A, SIDE_B, HEIGHT_B]</var>
<var id="ANGLE">atan(HEIGHT_A / SIDE_A) * 180 / PI</var>
<var id="TRI_A">new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {})</var>
<var id="TRI_B">function(){
var trA = new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50, {});
trA.labels = { "points" : ["A", "D", "E"] };
return trA;
}()
</var>
</div>
<p>
Given that:<br>
<code>
\quad\begin{eqnarray}
\overline{BC} &amp;=&amp; <var>HEIGHT_A</var> \\
\overline{BD} &amp;=&amp; <var>SIDE_B</var> \\
\overline{DE} &amp;=&amp; <var>HEIGHT_B</var> \\
\overline{AB} &amp;=&amp; x \\
\end{eqnarray}
</code>
</p>
<p class="question">What is the value of <code>x</code>?</p>
<div class="solution"><var>SIDE_A</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p><code>\overline{AB} : \overline{BC} = \dfrac{x}{<var>HEIGHT_A</var>}</code></p>
<p><code>\overline{AD} : \overline{DE} = \dfrac{x + <var>SIDE_B</var>}{<var>HEIGHT_B</var>}</code></p>
</div>
<p><code>
\dfrac{x}{<var>HEIGHT_A</var>} = \dfrac{x + <var>SIDE_B</var>}{<var>HEIGHT_B</var>}
</code></p>
<p><code>
<var>HEIGHT_B</var> \times x = <var>HEIGHT_A</var> \times (x + <var>SIDE_B</var>)
</code></p>
<p><code>
<var>HEIGHT_B</var>x = <var>coefficient(HEIGHT_A)</var>x + <var>HEIGHT_A * SIDE_B</var>
</code></p>
<p><code><var>plus(HEIGHT_B - HEIGHT_A + "x")</var> = <var>HEIGHT_A * SIDE_B</var></code></p>
<p data-if="HEIGHT_B - HEIGHT_A !== 1" ><code>
x = <var>fractionReduce(HEIGHT_A * SIDE_B, HEIGHT_B - HEIGHT_A)</var>
</code></p>
</div>
</div>
<div id="height-a">
<div class="vars">
<var id="SIDE_A">randRange(1, 10)</var>
<var id="SIDE_B">randRange(1, 10)</var>
<var id="HEIGHT_B">randRange(1, SIDE_A + SIDE_B)</var>
<var id="HEIGHT_A">HEIGHT_B * SIDE_A / (SIDE_A + SIDE_B)</var>
<var id="GCD">getGCD(SIDE_A + SIDE_B, HEIGHT_B)</var>
<var id="LABELS">[SIDE_A, "x", SIDE_B, HEIGHT_B]</var>
<var id="ANGLE">atan(HEIGHT_A / SIDE_A) * 180 / PI</var>
<var id="TRI_A">new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {})</var>
<var id="TRI_B">function(){
var trA = new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50, {});
trA.labels = { "points" : ["A", "D", "E"] };
return trA;
}()
</var>
</div>
<p>
Given that:<br>
<code>
\quad\begin{eqnarray}
\overline{AB} &amp;=&amp; <var>SIDE_A</var> \\
\overline{BD} &amp;=&amp; <var>SIDE_B</var> \\
\overline{DE} &amp;=&amp; <var>HEIGHT_B</var> \\
\overline{BC} &amp;=&amp; x
\end{eqnarray}
</code>
</p>
<p class="question">What is the value of <code>x</code>?</p>
<div class="solution"><var>HEIGHT_A</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p><code>\overline{AB} : \overline{BC} = \dfrac{<var>SIDE_A</var>}{x}</code></p>
<p><code>
\overline{AD} : \overline{DE} = \dfrac{<var>SIDE_A</var> + <var>SIDE_B</var>}{<var>HEIGHT_B</var>}
= <var>fraction(SIDE_A + SIDE_B, HEIGHT_B)</var>
<span data-if="GCD !== 1">= <var>fractionReduce(SIDE_A + SIDE_B, HEIGHT_B)</var></span>
</code></p>
</div>
<p><code>
\dfrac{<var>SIDE_A</var>}{x} = <var>fractionReduce(SIDE_A + SIDE_B, HEIGHT_B)</var>
</code></p>
<p><code>
<var>SIDE_A</var> <span data-if="HEIGHT_B !== GCD">\times <var>HEIGHT_B / GCD</var></span>
= <var>(SIDE_A + SIDE_B) / GCD</var> \times x
</code></p>
<p><code><var>SIDE_A * HEIGHT_B / GCD</var> = <var>coefficient((SIDE_A + SIDE_B) / GCD)</var>x</code></p>
<p><code>x = <var>fractionReduce(SIDE_A * HEIGHT_B, SIDE_A + SIDE_B)</var></code></p>
</div>
</div>
<div id="side-b">
<div class="vars" data-ensure="ANGLE > 20 && ANGLE < 50">
<var id="SIDE_A">randRange(1, 10)</var>
<var id="HEIGHT_A">randRange(1, SIDE_A)</var>
<var id="HEIGHT_B">HEIGHT_A + randRange(1, 10)</var>
<var id="SIDE_B">SIDE_A * (HEIGHT_B - HEIGHT_A) / HEIGHT_A</var>
<var id="GCD">getGCD(SIDE_A, HEIGHT_A)</var>
<var id="LABELS">[SIDE_A, HEIGHT_A, "x", HEIGHT_B]</var>
<var id="ANGLE">atan(HEIGHT_A / SIDE_A) * 180 / PI</var>
<var id="TRI_A">new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {})</var>
<var id="TRI_B">function(){
var trA = new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50, {});
trA.labels = { "points" : ["A", "D", "E"] };
return trA;
}()
</var>
</div>
<p>
Given that:<br>
<code>
\quad\begin{eqnarray}
\overline{AB} &amp;=&amp; <var>SIDE_A</var> \\
\overline{BC} &amp;=&amp; <var>HEIGHT_A</var> \\
\overline{DE} &amp;=&amp; <var>HEIGHT_B</var> \\
\overline{BD} &amp;=&amp; x
\end{eqnarray}
</code>
</p>
<p class="question">What is the value of <code>x</code>?</p>
<div class="solution"><var>SIDE_B</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p><code>
\overline{AB} : \overline{BC} = <var>fraction(SIDE_A, HEIGHT_A)</var>
<span data-if="GCD !== 1 || HEIGHT_A === 1">= <var>fractionReduce(SIDE_A, HEIGHT_A)</var></span>
</code></p>
<p><code>\overline{AD} : \overline{DE} = \dfrac{<var>SIDE_A</var> + x}{<var>HEIGHT_B</var>}</code></p>
</div>
<p><code>
\dfrac{<var>SIDE_A</var> + x}{<var>HEIGHT_B</var>} = <var>fractionReduce(SIDE_A, HEIGHT_A)</var>
</code></p>
<p><code>
<span data-if="HEIGHT_A !== GCD"><var>HEIGHT_A / GCD</var> \times (</span>
<var>SIDE_A</var> + x
<span data-if="HEIGHT_A !== GCD">)</span>
= <var>HEIGHT_B</var> \times <var>SIDE_A / GCD</var>
</code></p>
<p><code>
<var>HEIGHT_A * SIDE_A / GCD</var> + <var>coefficient(HEIGHT_A / GCD)</var>x = <var>HEIGHT_B * SIDE_A / GCD</var>
</code></p>
<p><code><var>coefficient(HEIGHT_A / GCD)</var>x = <var>(HEIGHT_B * SIDE_A - HEIGHT_A * SIDE_A) / GCD</var></code></p>
<p data-if="HEIGHT_A / GCD !== 1"><code>x = <var>fractionReduce(HEIGHT_B * SIDE_A - HEIGHT_A * SIDE_A, HEIGHT_A)</var></code></p>
</div>
</div>
<div id="height-b">
<div class="vars" data-ensure="ANGLE < 45">
<var id="SIDE_A">randRange(1, 10)</var>
<var id="SIDE_B">randRange(1, 10)</var>
<var id="HEIGHT_A">randRange(1, SIDE_A)</var>
<var id="HEIGHT_B">(SIDE_A + SIDE_B) * HEIGHT_A / SIDE_A</var>
<var id="GCD">getGCD(SIDE_A, HEIGHT_A)</var>
<var id="LABELS">[SIDE_A, HEIGHT_A, SIDE_B, "x"]</var>
<var id="ANGLE">atan(HEIGHT_A / SIDE_A) * 180 / PI</var>
<var id="TRI_A">new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {})</var>
<var id="TRI_B">function(){
var trA = new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50, {});
trA.labels = { "points" : ["A", "D", "E"] };
return trA;
}()
</var>
</div>
<p>
Given that:<br>
<code>
\quad\begin{eqnarray}
\overline{AB} &amp;=&amp; <var>SIDE_A</var> \\
\overline{BC} &amp;=&amp; <var>HEIGHT_A</var> \\
\overline{BD} &amp;=&amp; <var>SIDE_B</var> \\
\overline{DE} &amp;=&amp; x
\end{eqnarray}
</code>
</p>
<p class="question">What is the value of <code>x</code>?</p>
<div class="solution"><var>HEIGHT_B</var></div>
<div class="hints" data-apply="appendContents">
<div>
<p><code>
\overline{AB} : \overline{BC} = <var>fraction(SIDE_A, HEIGHT_A)</var>
<span data-if="GCD !== 1 || HEIGHT_A === 1">= <var>fractionReduce(SIDE_A, HEIGHT_A)</var></span>
</code></p>
<p><code>\overline{AD} : \overline{DE} = \dfrac{<var>SIDE_A</var> + <var>SIDE_B</var>}{x}
= \dfrac{<var>SIDE_A + SIDE_B</var>}{x}
</code></p>
</div>
<p><code><var>fractionReduce(SIDE_A, HEIGHT_A)</var> = \dfrac{<var>SIDE_A + SIDE_B</var>}{x}</code></p>
<p><code>
<var>SIDE_A / GCD</var> \times x =
<span data-if="HEIGHT_A !== GCD"><var>HEIGHT_A / GCD</var> \times </span>
<var>SIDE_A + SIDE_B</var>
</code></p>
<p><code>
<var>coefficient(SIDE_A / GCD)</var>x = <var>HEIGHT_A * (SIDE_A + SIDE_B) / GCD</var>
</code></p>
<p data-ensure="SIDE_A !== GCD"><code>
x = <var>fraction(HEIGHT_A * (SIDE_A + SIDE_B) / GCD, SIDE_A / GCD)</var>
<span data-if="getGCD(HEIGHT_A * (SIDE_A + SIDE_B) / GCD, SIDE_A / GCD) !== 1">
= <var>fractionReduce(HEIGHT_A * (SIDE_A + SIDE_B), SIDE_A)</var>
</span>
</code></p>
</div>
</div>
</div>
<div class="problem">
<div class="graphie" id="triangles">
var tri_range = TRI_B.boundingRange(1);
init({
range: [[tri_range[0][0], tri_range[0][1]],
[tri_range[1][0] - 0.2, tri_range[1][1]]],
scale: 400 / (tri_range[0][1] - tri_range[0][0])
})
style({
strokeWidth: 1,
stroke: KhanUtil.BLACK
});
var x_max = TRI_B.boundingRange()[0][1];
var y_max = TRI_B.boundingRange()[1][1];
var line_x = x_max * SIDE_A / (SIDE_A + SIDE_B);
var line_y = y_max * HEIGHT_A / HEIGHT_B;
var square = (tri_range[0][1] - tri_range[0][0]) / 36;
rect(x_max - square, 0, square, square);
rect(line_x - square, 0, square, square);
style({
strokeWidth: 2,
stroke: KhanUtil.BLUE
});
TRI_A.draw();
TRI_B.draw();
TRI_B.drawLabels();
style({
color: "black"
});
label([line_x, 0], "B", "below");
label([line_x, line_y], "C", "above");
label([line_x/2, 0], LABELS[0], "below");
label([line_x, line_y/2], LABELS[1], "right");
label([(SIDE_A + SIDE_B/2) * x_max / (SIDE_A + SIDE_B), 0], LABELS[2], "below");
label([x_max, y_max/2], LABELS[3], "right");
</div>
</div>
<div class="hints">
<p><code>\triangle ABC</code> and <code>\triangle ADE</code> both have a right angle and share <code>\angle BAC</code>.</p>
<p>Therefore <code>\triangle ABC</code> and <code>\triangle ADE</code> are similar triangles.</p>
<p>
Therefore, the ratio <code>\overline{AB} : \overline{BC}</code>
is equal to the ratio <code>\overline{AD} : \overline{DE}</code>.
</p>
</div>
</div>
</body>
</html>
Jump to Line
Something went wrong with that request. Please try again.