To determine which option has possible lengths for the other two sides of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths a, b, and c, the following conditions must hold:
- a + b > c
- a + c > b
- b + c > a
In this case, let's denote the two known sides as a and b, and the unknown side as x. According to the triangle inequality theorem, we can derive the following inequalities:
- \( a + b > x \)
- \( a + x > b \)
- \( b + x > a \)
We rearrange the inequalities to obtain bounds on x:
- \( x < a + b \) (1st inequality)
- \( x > |a - b| \) (2nd inequality)
Given that x is the length of the third side, it must be larger than the absolute difference of the two known sides and smaller than the sum of those two sides.
Now let's evaluate each of the provided options:
-
42 inches and 50 inches:
- \( x < 42 + 50 = 92 \)
- \( x > |42 - 50| = 8 \)
- So, x must be in the range \( 8 < x < 92 \).
-
28 inches and 64 inches:
- \( x < 28 + 64 = 92 \)
- \( x > |28 - 64| = 36 \)
- So, x must be in the range \( 36 < x < 92 \).
-
36 inches and 92 inches:
- \( x < 36 + 92 = 128 \)
- \( x > |36 - 92| = 56 \)
- So, x must be in the range \( 56 < x < 128 \).
-
48 inches and 76 inches:
- \( x < 48 + 76 = 124 \)
- \( x > |48 - 76| = 28 \)
- So, x must be in the range \( 28 < x < 124 \).
To summarize the ranges derived for each pair of side lengths:
- 42 and 50: \( 8 < x < 92 \)
- 28 and 64: \( 36 < x < 92 \)
- 36 and 92: \( 56 < x < 128 \)
- 48 and 76: \( 28 < x < 124 \)
Next, to find which of these pairs correspond to the range given for x (which we don't have the specifics of), you can cross-reference with the actual range indicated on the number line you mentioned.
Without the specific range of x, I can't definitively say which options are correct based purely on those calculations. Please provide the range for x shown on the number line to narrow it down further!