Find the length of the midsegment of the trapezoid?

3. The given figure looks like a right trapezium.

4. The given figure looks like a isosceles

trapezium. If so, then

3. Draw DE _|_ AB

Thus, MN=17+(21-17)/2=19.

4. Draw AE, DF _|_ BC

Thus MN=64+(82-64)/2=73.

The length of the midsegment of trapezoid is 

half the sum of the lengths of the two parallel sides.

3.

Midsegment = 19

4.

Midsegment = 73

First picture

2b.sin(β) = 21 - 17

2b.sin(β) = 4

b.sin(β) = 2

sin(β) = 2/b

ℓ = red + bleu

ℓ = 17 + b.sin(β) → we've just seen that: sin(β) = 2/b

ℓ = 17 + b.(2/b)

ℓ = 17 + 2

ℓ = 19

Second picture

white = 2a.sin(α) + 2b.sin(β) → you know that: white = 82 - 64

2a.sin(α) + 2b.sin(β) = 82 - 64

2a.sin(α) + 2b.sin(β) = 18

a.sin(α) + b.sin(β) = 9

a.sin(α) = 9 - b.sin(β)

ℓ = cyan + red + blue

ℓ = b.sin(β) + 64 + a.sin(α) → we've just seen that: a.sin(α) = 9 - b.sin(β)

ℓ = b.sin(β) + 64 + 9 - b.sin(β)

ℓ = 64 + 9

ℓ = 73

It's half the sum of the lengths of the parallel sides.  19 and 73 respectively.

It's the average of the parallel sides.

Q1. (21+17)/2 = 19

Q2. (64+82)/2 = 73