We have been given the following side lengths.
8 ft 12 ft
We can find the range of possible lengths for the third side of the triangle, x ft, using the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
XY+YZ>XZ
YZ+XZ>XY
XZ+XY>YZ
Applying this theorem to the given side lengths, we have three inequalities.
I:&29+51>x ⇒ 80 > x
II:&51+x>29 ⇒ x > -22
III:&x+29>51 ⇒ x > 22
The range for the possible lengths of the third side can be found by looking at the overlapping regions for these inequalities.
In interval notation, this can be written as the following compound inequality.
22< x< 80
Only side given in answer c. satisfies the inequalities above.
