(a) If point A (3, 4, 5) is the opposite vertex to the origin, then find the coordinates of the other six vertices of the box. (Order your answers from smallest to largest x, then from smallest to largest y, then from smallest to largest z.) (x, y, z)=
(x, y, z)=
(x, y, z)=
(x, y, z)=
(x, y, z)=
(x, y, z)=
(b) If point A (3, 4, 5) is the opposite vertex to the origin, then find the length of the diagonal of the box determined by the vertices O and A.

21

step-by-step explanation:

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set each factor equal to 0

4x-3 = 0

2x-1 = 0

solve for x.

4x-3=0

4x=3

x= 3/4

solve for x. 

2x-1= 0

2x= 1

x= 1/2

your answer should be 1/2 is less than or equal to x and 3/4 is greater than or equal to x

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4