Video: Applying Pythagoras’s Theorem to Solve More Complex Problems

𝐴𝐡𝐢𝐷𝐴′𝐡′𝐢′𝐷′ is a cube. Determine the lengths of line segment 𝐴′𝐡 and line segment 𝐴𝐢.

03:10

Video Transcript

𝐴 𝐡 𝐢 𝐷 𝐴 prime 𝐡 prime 𝐢 prime 𝐷 prime is a cube. Determine the lengths of line segment 𝐴 prime 𝐡 and line segment 𝐴𝐢.

The first thing we want to do is identify line segments 𝐴 prime 𝐡 and line segment 𝐴𝐢. Here is line segment 𝐴 prime 𝐡, and in yellow is line segment 𝐴𝐢. Line segment 𝐴 prime 𝐡 is part of the triangle 𝐴 𝐡 𝐴 prime. Because we know that this shape is a cube, each of its faces is a square. And that means that, on each face, the vertex is a right angle. It also means that every side length measures 97 centimeters.

What we see is that line segment 𝐴 prime 𝐡 is the hypotenuse of a right triangle. And that means that the length of 𝐴 prime 𝐡 squared would be equal to side 𝐴 𝐴 prime squared plus side 𝐴𝐡 squared.

We’re using the Pythagorean theorem. We know that the length of the hypotenuse squared is equal to the other two sides squared and then added together. 97 squared plus 97 squared equals 18818. Remember, that’s the squared length of our hypotenuse. So we take the square root.

We have to take the square root of 18818, but I want to rewrite 18818. Because I know that 18818 is 97 squared multiplied by two, we can break up the square root. We can say the square root of 97 squared times the square root of two. The square root of 97 squared is 97. And we’ll leave the square root of two as the square root of two. The measure of line segment 𝐴 prime 𝐡 equals 97 times the square root of two centimeters.

Now on to the length of line segment 𝐴𝐢, line segment 𝐴𝐢, what we notice here is that line segment 𝐴𝐢 is the hypotenuse of a triangle with side lengths 97 and 97. The triangle in the pink is the same size as the triangle in the yellow. And that means the length of line segment 𝐴𝐢 is equal to the lengths of line segment 𝐴 prime 𝐡, 97 times the square root of two centimeters.

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