10. The Pythagorean theoremhttps://peda.net/id/1c2007522cf2022-09-05T12:42:41+03:00https://peda.net/:static/533/peda.net.logo.bg.svg<div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div>
The Pythagorean theoremhttps://peda.net/id/1c230ff22cf2020-11-23T10:54:52+02:00<br/>
<span class="center small"><a href="https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/1pl/pythagoraan-lause/s2k0#top" title="10_Pythagorean-theorem.png"><img src="https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/1pl/pythagoraan-lause/s2k0:file/photo/6a475425c837d9d4fb67a9adb7ff1be4ff2973e8/10_Pythagorean-theorem.png" alt="" title="hypotenuse, leg, leg" class="inline" loading="lazy"/></a></span><br/>
<div class="eoppi-summary">
<p>In a <b>right triangle</b>, the <b>hypotenuse</b> is the side opposite to the <b>right angle</b>. Sides adjacent to the right angle are called <b>legs</b>.<br/>
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<b><span lang="EN-US">NB! </span></b><span lang="EN-US">The hypotenuse is always the longest side of a right triangle.</span></p>
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<p class="p1"><span class="left"><a href="https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/1pl/pythagoraan-lause/s2k02#top" title="Skärmavbild 2018-11-02 kl. 07.35.32.png"><img src="https://peda.net/qis/2022-2023/mathematics/matematiikka-8/oit/1pl/pythagoraan-lause/s2k02:file/photo/313eaabe1a4d83fa662f61f6c5df6d6c2d5de247/Ska%CC%88rmavbild%202018-11-02%20kl.%2007.35.32.png" alt="" title="Illustration of the Pythagorean theorem" class="inline" loading="lazy"/></a></span></p>
<div><br/>
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Consider a rectangular triangle with legs of [[$ 3 $]] and [[$ 4 $]] and a hypotenuse of [[$ 5 $]].</div>
<p><span>Draw squares on the sides of the triangle with side lengths that are equal to the triangle's side lengths.</span></p>
<p class="p1"><span lang="EN-US">The areas of the leg squares are<br/>
</span>[[$ 3^2 = 9 $]] and [[$ 4^2 = 16 $]].<br/>
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<span lang="EN-US">The area of the hypotenuse square is</span> [[$ 5^2 = 25 $]].</p>
<p class="p1">If the areas of the leg squares are added together [[$ 9 + 16 = 25 $]], the area of the hypotenuse square is obtained. <br/>
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This property is valid for all rectangular triangles and is known as <b>the Pythagorean theorem</b>.<br/>
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</p>
<div class="eoppi-summary">
<h3>The Pythagorean theorem</h3>
<p class="p1">In a <b>right triangle</b>, the sum of the squares of the two <b>legs</b> equals the square of the <b>hypotenuse</b>.<br/>
[[$$ a^2 + b^2 = c^2 $$]]</p>
</div>
<p><span>The Pythagorean theorem can be used to examine whether a triangle is a right triangle.</span></p>
<div class="eoppi-summary">
<h3>Examining whether a triangle is a right triangle</h3>
If the sum of the squares of the two shorter sides of a triangle is <b>equal</b> to the square of the longest side, the triangle is a right triangle.</div>
2022-09-05T12:42:41+03:00Exerciseshttps://peda.net/id/1c2466722cf2020-10-05T13:56:21+03:00<a href="https://peda.net/id/1c2b1c192cf" rel="noopener" target="_blank">Basic exercises<br/>
</a><a href="https://peda.net/id/1c2fffef2cf" rel="noopener" target="_blank">Applied exercises<br/>
</a><a href="https://peda.net/id/1c327ffb2cf" rel="noopener" target="_blank">Challenging exercises</a>2022-09-05T12:42:41+03:00