Question

1. what is the length of \\(\overline{ef}\\)? image of a trapezoid with ab = 22 in, dc = 27 in, and e, f as midpoints on ad, bc respectively options: 25 in, 49 in, 13.5 in, 24.5 in

Explanation:

Step1: Identify the theorem

This is a trapezoid with a midline (or midsegment) \( \overline{EF} \). The midline theorem for trapezoids states that the length of the midline is the average of the lengths of the two bases.

Step2: Apply the formula

The formula for the midline \( EF \) of a trapezoid is \( EF=\frac{AB + DC}{2} \), where \( AB = 22 \) in and \( DC = 27 \) in.
Substitute the values: \( EF=\frac{22 + 27}{2}=\frac{49}{2}=24.5 \) in? Wait, no, wait. Wait, the options have 25, 49, 13.5, 24.5. Wait, maybe I made a mistake. Wait, no, the figure: \( AB \) and \( DC \) are the two bases, and \( EF \) is the midline. Wait, but let's check again. Wait, the segments \( AE = ED \) and \( BF = FC \), so \( E \) and \( F \) are midpoints of the legs. Then the midline length is the average of the two bases. So \( (22 + 27)/2=(49)/2 = 24.5 \)? But wait, the option is 24.5? Wait, no, the options are 25, 49, 13.5, 24.5. Wait, maybe I miscalculated. Wait, 22 + 27 is 49, divided by 2 is 24.5. So that's one of the options. Wait, but let me check again. Wait, maybe the figure is a parallelogram? No, it's a trapezoid? Wait, no, if \( AE = ED \) and \( BF = FC \), and \( AB \) and \( DC \) are parallel, then \( EF \) is the midline. So the formula is correct. So \( (22 + 27)/2 = 24.5 \)? Wait, but 22 + 27 is 49, divided by 2 is 24.5. So the answer should be 24.5? Wait, but the option is 24.5? Wait, the options are: 25 in, 49 in, 13.5 in, 24.5 in. So 24.5 is an option. Wait, but maybe I made a mistake. Wait, no, let's recalculate: 22 + 27 = 49, 49 / 2 = 24.5. So that's correct.

Wait, but wait, maybe the figure is a trapezoid where \( AB \) and \( DC \) are the two bases, and \( EF \) is the midline. So the calculation is correct. So the length of \( \overline{EF} \) is 24.5 in? But wait, the option is 24.5? Wait, the options are:

• 25 in

• 49 in

• 13.5 in

• 24.5 in

So the correct answer is 24.5 in? Wait, but let me check again. Wait, maybe the problem is that \( AB \) and \( DC \) are the two bases, and \( EF \) is the midline. So the formula is \( EF = (AB + DC)/2 \). So (22 + 27)/2 = 24.5. So that's the answer.

Answer:

24.5 in (Option: 24.5 in)