WEBVTT
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In the figure, the line segment π΄π΅ lies in the plane π₯ and the line segment π΄πΆ is perpendicular to π₯.
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Given that π΄π΅ equals six and π΄πΆ equals eight, find the length of the line segment π΅πΆ.
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Letβs begin by adding what we know to our diagram.
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Weβre told that π΄π΅ is equal to six.
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So the line segment π΄π΅ must be six units in length.
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Similarly, π΄πΆ is equal to eight.
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So the line segment π΄πΆ is eight units.
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Weβre looking to find the length of the line segment π΅πΆ, as shown.
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So we need to use the final piece of information in our question.
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Weβre told that the line segment π΄π΅ lies in the plane π₯.
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And the line segment π΄πΆ is perpendicular to that same plane.
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This means the line segment π΄π΅ must be perpendicular to the line segment π΄πΆ.
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That is, π΄π΅ and π΄πΆ form a right angle.
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Iβve redrawn the right-angle triangle so we can see whatβs happening in a little bit more detail.
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Labelling the unknown side π, we see that we know the length of the short two sides in the right-angle triangle.
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And π is the hypotenuse.
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We can therefore use the Pythagorean theorem to work out the length of the missing side.
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This says that the sum of the squares of the short two sides must be equal to the square of the longest side.
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Itβs usually written as π squared plus π squared equals π squared.
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Of course, in our triangle, the longest side is labelled π.
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So we could say that π squared plus π squared equals π squared.
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Substituting the values from our triangle in, and we find that eight squared plus six squared equals π squared.
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Now, you might recognize a Pythagorean triple hit.
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But weβre going to solve the equation just in case.
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Eight squared is 64.
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And six squared is 36.
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Their sum is 100.
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So we find π squared is equal to 100.
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We solve this equation by finding the square root of 100.
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Remember, we donβt need to find both a positive and negative square root since weβre dealing with a length.
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It absolutely must be positive.
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And when we find the square root of 100, we get 10.
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The length of π΅πΆ is 10 units.
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Now, in fact, six, eight, and 10 form a Pythagorean triple.
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We know that six squared plus eight squared equals 10 squared.
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So we couldβve deduced quite quickly that π was equal to 10.
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It is, of course, absolutely fine to solve the equation.