Which of the following is the statement of the triangle inequality theorem?
a. the sum of the measures of any two angles of a triangle is greater than the measure of the third angle.
b. the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
c. the longest side of a triangle is opposite the largest angle.
d. the largest angle of a triangle is between the two longest sides.

12*c

step-by-step explanation:

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