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<!DOCTYPE html>
<html data-require="math graphie math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Special right triangles</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
<script>
function betterTriangle(width, height, A, B, C, a, b, c) {
var scale = 5 / Math.sqrt(width * width + height * height);
width *= scale;
height *= scale;
with ( KhanUtil.currentGraph ) {
// Leave some space for the labels
init({ range: [[-1.5, width + 1], [-1, height + 1]] });
path([ [0, 0], [width, 0], [0, height], true ]);
label( [0, height], A, "above left" );
label( [0, 0], C, "below left" );
label( [width, 0], B, "below right" );
label( [0, height/2], b, "left" );
label( [width/2, 0], a, "below" );
label( [width/2, height/2], c, "above right", {
labelDistance: 3
} );
}
}
</script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="45-45-90-find-hypotenuse">
<div class="vars">
<var id="AC">randRange(2, 7)</var>
</div>
<div class="question">
<p>In the right triangle shown, <code>AC = BC = <var>AC</var></code>.</p>
<p>What is <code>AB</code>?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle( 1, 1, "A", "B", "C", AC, AC, "x" );
</div>
</div>
<div class="solution" data-type="radical"><var>AC * AC * 2</var></div>
<div class="hints">
<p>
We know the length of each leg, and want to find the length of the hypotenuse. What
mathematical relationship is there between a right triangle's leg and its hypotenuse?
</p>
<p><code>\qquad AC^2 + BC^2 = x^2</code></p>
<p><code>\qquad <var>AC</var>^2 + <var>AC</var>^2 = x^2</code></p>
<p><code>\qquad x^2 = 2 \cdot <var>AC</var>^2</code></p>
<p><code>\qquad x = \sqrt{2 \cdot <var>AC</var>^2}</code></p>
<p><code>\qquad x = \sqrt{2} \cdot \sqrt{<var>AC</var>^2}</code></p>
<p><code>\qquad x = <var>AC</var>\sqrt{2}</code></p>
</div>
</div>
<div id="45-45-90-find-leg">
<div class="vars">
<var id="AB">2 * randRange(2, 6)</var>
</div>
<div class="question">
<p>In the right triangle shown, <code>AC = BC</code> and <code>AB = <var>AB</var></code>.</p>
<p>How long are each of the legs?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle( 1, 1, "A", "B", "C", "x", "x", AB );
</div>
</div>
<div class="solution" data-type="radical"><var>AB * AB / 2</var></div>
<div class="hints">
<p>
We know the length of the hypotenuse, and want to find the length of each leg.
What mathematical relationship is there between a right triangle's legs and its hypotenuse?
</p>
<p><code>\qquad x^2 + x^2 = AB^2</code></p>
<p><code>\qquad 2 \cdot x^2 = <var>AB</var>^2</code></p>
<p><code>\qquad x^2 = <var>AB * AB / 2</var></code></p>
<p><code>\qquad x = \sqrt{<var>AB * AB / 2</var>}</code></p>
<p><code>\qquad x = \sqrt{<var>AB * AB / 4</var> \cdot 2}</code></p>
<p><code>\qquad x = \sqrt{<var>AB * AB / 4</var>} \cdot \sqrt{2}</code></p>
<p><code>\qquad x = <var>AB/2</var> \sqrt{2}</code></p>
</div>
</div>
<div id="45-45-90-find-leg-2">
<div class="vars">
<var id="AB">2 * randRange(2, 6)</var>
</div>
<div class="question">
<p>In the right triangle shown, <code>AC = BC</code> and <code>AB = <var>AB</var>\sqrt{2}</code>.</p>
<p>How long are each of the legs?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle( 1, 1, "A", "B", "C", "x", "x", AB + "\\sqrt{2}" );
</div>
</div>
<div class="solution" data-type="radical"><var>AB * AB</var></div>
<div class="hints">
<p>
We know the length of the hypotenuse, and want to find the length of each leg.
What mathematical relationship is there between a right triangle's legs and its hypotenuse?
</p>
<p><code>\qquad x^2 + x^2 = AB^2</code></p>
<p><code>\qquad 2 \cdot x^2 = (<var>AB</var>\sqrt{2})^2</code></p>
<p><code>\qquad 2 \cdot x^2 = <var>AB</var>^2 \cdot (\sqrt{2})^2</code></p>
<p><code>\qquad 2 \cdot x^2 = <var>AB * AB</var> \cdot 2</code></p>
<p><code>\qquad x^2 = <var>AB * AB</var></code></p>
<p><code>\qquad x = <var>AB</var></code></p>
</div>
</div>
<div id="30-60-90-find-hypotenuse-given-short-leg" data-weight="2">
<div class="vars">
<var id="BC">randRange(2, 6)</var>
<var id="BCr, BCrs">randFromArray([[1, ""], [3, "\\sqrt{3}"]])</var>
<var id="BCdisp">BC + BCrs</var>
<var id="ANGLE">rand(2)</var>
<var id="mAB">["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]</var>
</div>
<div class="question">
<p>In the right triangle shown, <code><var>mAB</var></code> and <code>BC = <var>BC + BCrs</var></code>.</p>
<p>How long is <code>AB</code>?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", BC + BCrs, "", "x");
if (ANGLE === 0) {
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
} else {
arc([5/2,0], 0.5, 120, 180);
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
}
</div>
</div>
<div class="solution" data-type="radical"><var>4 * BC * BC * BCr</var></div>
<div class="hints">
<p>
We know the length of a leg, and want to find the length of the hypotenuse.
What is the relationship between the sides of a <code>30 - 60 - 90</code> triangle?
</p>
<div>
<p>This is a <code>30-60-90</code> triangle with a hypotenuse of length <code>1</code>.</p>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1);
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
arc([5/2,0], 0.5, 120, 180);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
</div>
</div>
<p>The ratio of <code>AB : BC</code> is <code>1 : \dfrac{1}{2}</code>.</p>
<p>Therefore, <code>\dfrac{x}{<var>BC + BCrs</var>} = \dfrac{1}{\frac{1}{2}} = 2</code>.</p>
<p><code>x = 2 \cdot <var>BC + BCrs</var></code></p>
<p><code>x = <var>BC * 2 + BCrs</var></code></p>
</div>
</div>
<div id="30-60-90-find-hypotenuse-given-long-leg" data-weight="2">
<div class="vars">
<var id="AC">3 * randRange( 2, 6 )</var>
<var id="ACr, ACrs, ABs, AB">randFromArray([
[1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3],
[3, "\\sqrt{3}", (AC * 2), AC * AC * 4]
])</var>
<var id="ACdisp">AC + ACrs</var>
<var id="ANGLE">rand(2)</var>
<var id="mAB">["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]</var>
</div>
<div class="question">
<p>In the right triangle shown, <code><var>mAB</var></code> and <code>AC = <var>AC + ACrs</var></code>.</p>
<p>How long is <code>AB</code>?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "", AC + ACrs, "x");
if (ANGLE === 0) {
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
} else {
arc([5/2,0], 0.5, 120, 180);
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
}
</div>
</div>
<div class="solution" data-type="radical"><var>AB</var></div>
<div class="hints">
<p>
We know the length of a leg, and want to find the length of the hypotenuse.
What is the relationship between the sides of a <code>30 - 60 - 90</code> triangle?
</p>
<div>
<p>This is a <code>30-60-90</code> triangle with a hypotenuse of length <code>1</code>.</p>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1);
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
arc([5 / 2,0], 0.5, 120, 180);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
label([5 / 2 - 0.2, 0], "{60}^{\\circ}", "above left");
</div>
</div>
<p>The ratio of <code>AB : AC</code> is <code>1 : \dfrac{\sqrt{3}}{2}</code>.</p>
<p>Therefore, <code>\dfrac{x}{<var>AC + ACrs</var>} = \dfrac{1}{\frac{\sqrt{3}}{2}} = \dfrac{2}{\sqrt{3}}</code>.</p>
<p><code>x = \dfrac{2}{\sqrt{3}} \cdot <var>AC + ACrs</var></code></p>
<div data-if="ACrs === ''">
<p><code>x = \dfrac{<var>2 * AC</var>}{\sqrt{3}}</code></p>
<p><code>x = \dfrac{<var>2 * AC</var>}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}}</code></p>
<p><code>x = \dfrac{<var>2 * AC</var> \cdot \sqrt{3}}{3}</code></p>
</div>
<p>So, <code>x = <var>ABs</var></code>.</p>
</div>
</div>
<div id="30-60-90-find-short-leg-given-hypotenuse" data-weight="2">
<div class="vars">
<var id="BC">randRange( 2, 6 )</var>
<var id="BCr, BCrs">randFromArray([ [1, ""], [3, "\\sqrt{3}"] ])</var>
<var id="ABdisp">2*BC + BCrs</var>
<var id="ANGLE">rand(2)</var>
<var id="mAB">["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]</var>
</div>
<div class="question">
<p>
In the right triangle shown, <code><var>mAB</var></code> and <code>AB = <var>2 * BC + BCrs</var></code>.
</p>
<p>
How long is <code>BC</code>?
</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "x", "", 2 * BC + BCrs);
if (ANGLE === 0) {
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
} else {
arc([5/2,0], 0.5, 120, 180);
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
}
</div>
</div>
<div class="solution" data-type="radical"><var>BC * BC * BCr</var></div>
<div class="hints">
<p>
We know the length of the hypotenuse, and want to find the length of the shortest side.
What is the relationship between the sides of a <code>30 - 60 - 90</code> triangle?
</p>
<div>
<p>This is a <code>30-60-90</code> triangle with a hypotenuse of length <code>1</code>.</p>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1);
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
arc([5/2,0], 0.5, 120, 180);
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
</div>
</div>
<p>The ratio of <code>BC : AB</code> is <code>\dfrac{1}{2} : 1</code>.</p>
<p>Therefore, <code>\dfrac{x}{<var>ABdisp</var>} = \dfrac{1}{2}</code>.</p>
<p><code>x = <var>BC + BCrs</var></code></p>
</div>
</div>
<div id="30-60-90-find-long-leg-given-hypotenuse" data-weight="2">
<div class="vars">
<var id="AC">3 * randRange( 2, 6 )</var>
<var id="ACr, ACrs, ABs, AB">randFromArray([
[1, "", (AC * 2 / 3) + "\\sqrt{3}", AC * AC * 4 / 3],
[3, "\\sqrt{3}", (AC * 2), AC * AC * 4]
])</var>
<var id="ANGLE">rand(2)</var>
<var id="mAB">["\\angle A = 30^\\circ", "\\angle B = 60^\\circ"][ANGLE]</var>
</div>
<div class="question">
<p>In the right triangle shown, <code><var>mAB</var></code> and <code>AB = <var>ABs</var></code>.</p>
<p>How long is <code>AC</code>?</p>
<div class="render-answer-area-here"></div>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "", "x", ABs);
if (ANGLE === 0) {
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
} else {
arc([5/2,0], 0.5, 120, 180);
label([5/2-0.2, 0], "{60}^{\\circ}", "above left");
}
</div>
</div>
<div class="solution" data-type="radical"><var>AC * AC * ACr</var></div>
<div class="hints">
<p>
We know the length of the hypotenuse, and want to find the length of the longest side.
What is the relationship between the sides of a <code>30 - 60 - 90</code> triangle?
</p>
<div>
<p>This is a <code>30-60-90</code> triangle with a hypotenuse of length <code>1</code>.</p>
<div class="graphie">
betterTriangle(1, sqrt(3), "A", "B", "C", "\\dfrac{1}{2}", "\\dfrac{\\sqrt{3}}{2}", 1);
arc([0, 5 * sqrt(3) / 2], 0.8, 270, 300);
arc([5 / 2,0], 0.5, 120, 180);
label([-0.1, (5 * sqrt(3) / 2) - 1], "{30}^{\\circ}", "below right");
label([5 / 2 - 0.2, 0], "{60}^{\\circ}", "above left");
</div>
</div>
<p>The ratio of <code>AC : AB</code> is <code>\dfrac{\sqrt{3}}{2} : 1</code>.</p>
<p>Therefore, <code>\dfrac{x}{<var>ABs</var>} = \dfrac{\sqrt{3}}{2}</code>.</p>
<p><code>x = \dfrac{\sqrt{3}}{2} \cdot <var>ABs</var></code></p>
<p data-if="ACrs === ''"><code>x = <var>AC / 3</var> \cdot \sqrt{3} \cdot \sqrt{3}</code></p>
<p>So, <code>x = <var>AC + ACrs</var></code>.</p>
</div>
</div>
</div>
</div>
</body>
</html>