Triangle Inequality Theorem: Proof that 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 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 C and D, we can use the triangle inequality theorem.

The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides is always greater than the length of the third side.

In this case, we can consider triangle CMD. CD is one side of the triangle, CM is another side, and MD is the remaining side.

According to the triangle inequality theorem, CD + CM > MD and CD + MD > CM.

Now, let’s assume that CD is not greater than either CM or MD. If CD was not greater than CM, then CM would be greater than CD. So, we can write this inequality as CM > CD.

Similarly, if CD was not greater than MD, then we can write this inequality as MD > CD.

Now, let’s add these two inequalities together:

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

Since CM + MD is greater than 2CD, it means that CD is less than half the sum of CM and MD.

However, according to the triangle inequality theorem, we know that CM + MD is always greater than CD. Therefore, our previous assumption that CD is not greater than either CM or MD is incorrect.

In conclusion, CD is always greater than either CM or MD when point M is between C and D.

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