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# if the three segments below were used to form a triangle, what could the values for the length of the third side be? check all that apply.

Line AC = 3
Line CB =12
Line BA =?

A) 3
B) 6
C) 9
D) 11
E) 12
F) 16

Make believe you have a 12 inch stick and a 3 inch stick nailed together at one end (loose enough to be able to pivot).

How big a stick would you need to connect the other ends of the 12 and 3 inch sticks.

If the 3 inch stick is laying on top of the 12 inch stick you'd need just a 9 inch stick. But to make the combination a triangle the third stick would have to be a little bigger than 9 inches.

If the 3 inch stick is rotated 180º so it's collinear with the 12 inch stick but not laying on top of it, you'd now need a 15 inch stick to connect the ends. But to make the combination a triangle the third stick would have to be a little smaller than 15 inches.

So the third stick's length has to be constrained to: 9 < BA < 15.

Both D and E satisfy the constraints.
Hi Claudia;
I am assuming that BA is the hypotenuse opposite a 90-degree angle.
AC=3
CB=12
a2+b2=c2
32+122=c2
9+144=c2
153=c2
√153=c
12.4=c
12=c

THE ANSWER IS E, IF MY ASSUMPTIONS ARE CORRECT.
Rule for Δ ABC: any two sides added together must be greater than the third side.
If
AB = c
BC = a
AC = b,
then
a + b > c
a + c > b
b + c > a
~~~~~~~~~~~~

a = 12
b = 3
c = ?

1.)  12 + 3 > c

15 > 3, 6, 9 , 11,  12.

(A, B, C, D, E)

2.)  12 + c > 3

12 + (all choices) > 3

3.)  3 + c > 12

3 + 11 > 12   (D)
3 + 12 > 12   (E)

Thus, choices (D) and (E) satisfy all three conditions.