Discussion
Which of the following can be the length of the third side of a triangle whose two sides are of lengths 19 cm and 15 cm?
Answer: C
The third side can be 33 cm.
By the triangle inequality, each side must be less than the sum and greater than the difference of the other two:
\[ 19 - 15 \lt c \lt 19 + 15,\ \text{so}\ 4 \lt c \lt 34 \]
Only 33 cm lies strictly between 4 cm and 34 cm. A side of 4 cm or 34 cm would make the three points lie on one straight line, so no triangle forms, and 3 cm is too short to close the triangle.
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