To determine the possible lengths of $AC$ in triangle $ABC$ with given sides $AB = 42$ and $BC = 20$, we can apply the Triangle Inequality Theorem. This theorem states that in any triangle, the length of any side must be less than the sum and greater than the difference of the other two sides.

So, for triangle $ABC$:

1. $AB + BC > AC$
2. $AB + AC > BC$
3. $BC + AC > AB$

Since we are looking for possible lengths of $AC$, we focus on the inequality: $|AB - BC| < AC < AB + BC$

Step-by-Step Calculation:

Given:

• $AB = 42$
• $BC = 20$

1. Calculate the Lower Bound: $|AB - BC| = |42 - 20| = 22$ So, $AC > 22$.

2. Calculate the Upper Bound: $AB + BC = 42 + 20 = 62$ So, $AC < 62$.

Possible Lengths for $AC$:

From the above, the possible values of $AC$ must satisfy $22 < AC < 62$.

The choices given are: 12, 20, 22, 32, 42, 50, 62, and 70.

The values that fall within the range $22 < AC < 62$ are:

• 32
• 42
• 50

Thus, the correct answers are 32, 42, and 50.

Would you like a deeper explanation or have any questions?

Here are five related questions that build on this information:

1. How does the Triangle Inequality Theorem apply to non-triangular shapes?
2. What would happen if one side of a triangle equaled the sum of the other two sides?
3. Could a triangle exist with side lengths 42, 20, and 62? Why or why not?
4. How can we generalize the Triangle Inequality for an obtuse triangle?
5. What other geometric properties can help determine the lengths of sides in a triangle?

Tip: Remember, the Triangle Inequality Theorem always holds, no matter the type of triangle (scalene, isosceles, or equilateral).