Question Video: Using the Pythagorean Theorem to Find an Unknown Side Length | Nagwa Question Video: Using the Pythagorean Theorem to Find an Unknown Side Length | Nagwa

# Question Video: Using the Pythagorean Theorem to Find an Unknown Side Length Mathematics • Second Year of Secondary School

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In the figure, the line segment π΄π΅ lies in the plane π₯ and the line segment π΄πΆ is perpendicular to π₯. Given that π΄π΅ = 6 and π΄πΆ = 8, find the length of the line segment π΅πΆ.

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### Video Transcript

In the figure, the line segment π΄π΅ lies in the plane π₯ and the line segment π΄πΆ is perpendicular to π₯. Given that π΄π΅ equals six and π΄πΆ equals eight, find the length of the line segment π΅πΆ.

Letβs begin by adding what we know to our diagram. Weβre told that π΄π΅ is equal to six. So the line segment π΄π΅ must be six units in length. Similarly, π΄πΆ is equal to eight. So the line segment π΄πΆ is eight units. Weβre looking to find the length of the line segment π΅πΆ, as shown. So we need to use the final piece of information in our question. Weβre told that the line segment π΄π΅ lies in the plane π₯. And the line segment π΄πΆ is perpendicular to that same plane. This means the line segment π΄π΅ must be perpendicular to the line segment π΄πΆ. That is, π΄π΅ and π΄πΆ form a right angle.

Iβve redrawn the right-angle triangle so we can see whatβs happening in a little bit more detail. Labelling the unknown side π, we see that we know the length of the short two sides in the right-angle triangle. And π is the hypotenuse. We can therefore use the Pythagorean theorem to work out the length of the missing side. This says that the sum of the squares of the short two sides must be equal to the square of the longest side. Itβs usually written as π squared plus π squared equals π squared. Of course, in our triangle, the longest side is labelled π. So we could say that π squared plus π squared equals π squared. Substituting the values from our triangle in, and we find that eight squared plus six squared equals π squared.

Now, you might recognize a Pythagorean triple hit. But weβre going to solve the equation just in case. Eight squared is 64. And six squared is 36. Their sum is 100. So we find π squared is equal to 100. We solve this equation by finding the square root of 100. Remember, we donβt need to find both a positive and negative square root since weβre dealing with a length. It absolutely must be positive. And when we find the square root of 100, we get 10. The length of π΅πΆ is 10 units. Now, in fact, six, eight, and 10 form a Pythagorean triple. We know that six squared plus eight squared equals 10 squared. So we couldβve deduced quite quickly that π was equal to 10. It is, of course, absolutely fine to solve the equation.

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