𝐴 𝐵 𝐶 𝐷 𝐴 prime 𝐵 prime 𝐶
prime 𝐷 prime is a cube. Determine the lengths of line
segment 𝐴 prime 𝐵 and line segment 𝐴𝐶.
The first thing we want to do is
identify line segments 𝐴 prime 𝐵 and line segment 𝐴𝐶. Here is line segment 𝐴 prime 𝐵,
and in yellow is line segment 𝐴𝐶. Line segment 𝐴 prime 𝐵 is part of
the triangle 𝐴 𝐵 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 𝐵 is the hypotenuse of a right triangle. And that means that the length of
𝐴 prime 𝐵 squared would be equal to side 𝐴 𝐴 prime squared plus side 𝐴𝐵
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 𝐵 equals 97 times the square root of two centimeters.
Now on to the length of line
segment 𝐴𝐶, line segment 𝐴𝐶, what we notice here is that line segment 𝐴𝐶 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 𝐴𝐶 is equal to the lengths of line segment 𝐴 prime 𝐵, 97 times the
square root of two centimeters.