Iba constructed a triangle ABC in her notebook. AB was 12 centimeters long. BC was 10 centimeters long. Select all measurements that could be the length of AC. a) 2cm b) 3cm c) 8cm d) 16cm e) 22cm f) 25cm (NOTE: All possible answers (2 cm, 3 cm, 8 cm, 16 cm, 22 cm).

At first we should know that:

" The sum of the lengths of two sides of a triangle must be grater than the length of the length of the third side "
which mean:
For a triangle ABC :
                                AB + BC > AC
                           &   AB + AC > BC
                           &   AC + BC > AB
Applying the theorem to the problem:
we have AB = 12 cm. and BC = 10 cm.

For option (a): if AC = 2 cm. ⇒⇒⇒   AC + BC = 10 + 2 = 12 = AB
                         So option (a)   ⇒ could not be the length of AC

For option (b): if AC = 3 cm. ⇒⇒⇒   AB + BC = 12 + 10 = 22 > AC
                                                   &   AB + AC = 12 + 3 = 15 > BC
                         &   AC + BC = 3 + 10 = 13 > AB
                         So option (b)   ⇒ could be the length of AC

For option (c): if AC = 8 cm. ⇒⇒⇒   AB + BC = 12 + 10 = 22 > AC
                                                   &   AB + AC = 12 + 8 = 20 > BC
                         &   AC + BC = 8 + 10 = 18 > AB
                         So option (c)   ⇒ could be the length of AC

For option (d): if AC = 16 cm. ⇒⇒⇒   AB + BC = 12 + 10 = 22 > AC
                                                   &   AB + AC = 12 + 16 = 28 > BC
                         &   AC + BC = 16 + 10 = 26 > AB
                         So option (d)   ⇒ could be the length of AC

For option (e): if AC = 22 cm. ⇒⇒⇒   AB + BC = 10 + 12 = 22 = AC
                         So option (e)   ⇒ could not be the length of AC

For option (f): if AC = 25 cm. ⇒⇒⇒   AB + BC = 12 + 10 = 22 < AC
                         So option (f)   ⇒ could not be the length of AC

The measurements that could be the length of AC are 3 , 8 , 16

i.e.: options (b),(c) and (d)