The Triangle Inequality Theorem: Proving CD is Greater than CM or MD

If point M is between points C and D, then CD is greater than either CM or MD

To prove that CD is greater than either CM or MD when point M is between points C and D, we can use the Triangle Inequality Theorem

To prove that CD is greater than either CM or MD when point M is between points C and D, we can use the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that for any triangle with sides a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side.

In this case, we can consider the triangle CMD. CD represents the side opposite to angle M, CM represents the side opposite to angle D, and MD represents the side opposite to angle C.

According to the Triangle Inequality Theorem, we have:

CD + CM > MD (1)
CD + MD > CM (2)
CM + MD > CD (3)

From equations (1) and (2), we can add both sides together to get:

CD + CM + CD + MD > MD + CM
2CD > MD + CM

Similarly, from equations (1) and (3), we can add both sides together to get:

CD + CM + CM + MD > CD + MD
2CM > CD + MD

Finally, adding the two inequalities together, we have:

2CD + 2CM > MD + CM + CD + MD
2(CD + CM) > 2(CM + MD)
CD + CM > CM + MD

Therefore, we can conclude that CD is greater than either CM or MD when point M is between points C and D.

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