Question

If DE = 37 cm and EF = 16 cm, then what are the possible lengths for DF so that DE,EF, and DF can form a triangle? Explain your reasoning.

Answer 1

The length of DF must be between 21 and 53.

Explanation:

In a triangle, the length of two sides added together must exceed the length of the 3rd side. So, since EF is the shortest of the two givens, we know that EF + DF must be greater than DE. So we can plug in these numbers to find the minimum.

EF + DF > DE

16 + DF > 37

DF > 21

Now, for the upper maximum, we know that the two given lengths must be greater than the length of DF. So again, we can solve for the maximum using the amounts.

DE + EF > DF

37 + 16 > DF

53 > DF

With these two in mind, we know that DF must be between 21 and 53

Answer 2

Answer: 21 < DF < 53 ; Triangle Inequality Theorem

Explanation:

The Triangle Inequality Theorem states that the sum of any two sides is greater than the sum of the third side.

DE + EF > DF EF + DF > DE DE + DF > EF

37 + 16 > DF 16 + DF > 37 37 + DF > 16

53 > DF DF > 21 DF > -21

Now let's put together the answers. Note that DF > -21 is irrelevant since all lengths must be greater than 0.

53 > DF and DF > 21 means that DF is between 53 and 21

-----> 21 < DF < 53