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Maths

Maths

3D Pythagoras - Higher

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3D Pythagoras

In a nutshell

So far, you have used Pythagoras' theorem to find lengths of 22​ dimensional shapes. However, Pythagoras' theorem can also applied to find distances in 33​ dimensions.



Using Pythagoras' theorem in 33​ dimensions

It is best to show this with a sketch. Here is a cuboid ABCDEFGHABCDEFGH​:



To find the distance AFAF​ (=d=d), first use Pythagoras' theorem on triangle ABHABH​ to find the distance AHAH​, and then use it again on triangle AFHAFH​ to find the distance AFAF​.

Example 1

In the cuboid shown above, a=3cm,b=4cm,d=13cma=3cm,b=4cm,d=13cm. Find the value of cc.


Find AHAH by focusing on the right-angled triangle ABHABH:

a2+b2=(AH)2a^2+b^2=(AH)^2​​​​

32+42=(AH)23^2+4^2=(AH)^2​​

(AH)2=25(AH)^2=25​​

AH=25=5cmAH=\sqrt{25}=5cm​​


Then, find cc by focusing