A rectangular prism has a length of 8, width of 9, and height of 12. What is the length of its longest internal diagonal?

AdminMod2
Sep 16, 2017

#1**+1 **

The longest diagonal will be from one corner to the corner completely opposite it (top and bottom, front and back, left and right). Like the black line in this picture:

We can find it's length by making it into a triangle like this:

Let's call this new gray line " b ".

Now we can use the Pythagorean theorem to find b and the length of the prism's diagonal.

9^{2} + 8^{2} = b^{2}

81 + 64 = b^{2}

145 = b^{2}

b^{2} + 12^{2} = (diagonal)^{2} Substitute 145 in for b^{2} .

145 + 12^{2} = (diagonal)^{2}

145 + 144 = (diagonal)^{2}

289 = (diagonal)^{2} Take the positive square root of both sides.

√289 = diagonal

17 = diagonal

hectictar
Sep 16, 2017

#1**+1 **

Best Answer

The longest diagonal will be from one corner to the corner completely opposite it (top and bottom, front and back, left and right). Like the black line in this picture:

We can find it's length by making it into a triangle like this:

Let's call this new gray line " b ".

Now we can use the Pythagorean theorem to find b and the length of the prism's diagonal.

9^{2} + 8^{2} = b^{2}

81 + 64 = b^{2}

145 = b^{2}

b^{2} + 12^{2} = (diagonal)^{2} Substitute 145 in for b^{2} .

145 + 12^{2} = (diagonal)^{2}

145 + 144 = (diagonal)^{2}

289 = (diagonal)^{2} Take the positive square root of both sides.

√289 = diagonal

17 = diagonal

hectictar
Sep 16, 2017