Answer:
Midsegment is 35.
Explanation:
TL; DR:
The smaller triangle (with a base of 6x + 5) has side lengths that are twice as small as the entire triangle (the one with a base of 3x+55). So, 2(6x+5) = 3x + 55. x = 5, so the midsegment is 35.
Look at the photo attached below.
Here is the statement/reason:
1. ED ║ AB (Midsegments are parallel to the base)
Note: Then, CA and CB are transversals
2. Then, m∠CED = m∠EAB (corresponding angles)
3. m∠CDE = m∠DBA (corresponding angles)
4. ΔCED is similar to ΔCAB (angle angle)
5. CA = CE + EA
6. CA = 39 + 39 = 78 (Substitution, algebra)
7. ED/AB = CE/CA (similar triangles)
Note: You can also use CD and CB
8. (6x+5) / (3x+55) = 39 / 78 (substitution)
9. (6x + 5) / (3x + 55) = 1 / 2 (algebra)
10. 12x + 10 = 3x + 55, x = 5 (algebra)
11. ED = 6x + 5 (given)
12. ED = 6*5 + 5 = 35 (substitution, algebra)
An explanation of the algebra:
(6x+5) is twice as small as (3x + 55). So, we can write this equation:
2(6x+5) = 3x + 55
12x + 10 = 3x + 55
Subtract 3x from both sides.
9x + 10 = 55
Subtract 10 from both sides.
9x = 45
x = 5.
Now, put that into ED.
6x + 5 = 6 * 5 + 5 = 35
I hope this helps! Feel free to ask any questions!