A. find the length of the midsegment of an equilateral triangle with side lengths of 12.5 cm.
b. given that ut is the perpendicular bisector of ab, where t is on ab, find the length of at given at = 3x + 6 and tb = 42 - x.
c. given angle abc has angle bisector bd, where ab = cb, find the value of x if ad = 5x + 10 and dc = 28 - x.

a) The length of the mid-segment  is 6.25 cm

b) The length of AT = 33 units

c) The value of x is 3

Step-by-step explanation:

a)

* Lets explain the mid-segment of a triangle

- A mid-segment of a triangle is a segment connecting the midpoints

 of two sides of a triangle

- This segment has two special properties

# It is parallel to the third side

# The length of the mid-segment is half the length of the third side

∵ The triangle is equilateral triangle

∴ All sides are equal in length

∵ the side lengths = 12.5 cm

∵ The length of the mid-segment = 1/2 the length of the third side

∴ The length of the mid-segment = 1/2 × 12.5 = 6.25 cm

* The length of the mid-segment  is 6.25 cm

b)

∵ UT is a perpendicular bisector of AB

∵ T lies on AB

∴ T is the mid-point of AB

∵ AT = BT

∵ AT = 3x + 6

∵ BT = 42 - x

- Equate AT and BT

∴ 3x + 6 = 42 - x

- Add x to both sides

∴ 4x + 6 = 42

- Subtract 6 from both sides

∴ 4x = 36

- Divide both sides by 4

∴ x = 9

∵ AT = 3x + 6

- Substitute x by 9

∴ AT = 3(9) + 6 = 27 + 6 = 33

* The length of AT = 33 units

c)

- In Δ ABC

∵ AB = BC

∴ Δ ABC is an isosceles triangle

∵ BD bisects angle ABC

- In the isosceles Δ the bisector of the vertex angle bisects the base

 of the triangle which is opposite to the vertex angle

∵ AC is the opposite side of the vertex B

∴ BD bisects the side AC at D

∴ AD = CD

∵ AD = 5x + 10

∵ CD = 28 - x

∴ 5x + 10 = 28 - x

- Add x to both sides

∴ 6x + 10 = 28

- Subtract 10 from both sides

∴ 6x = 18

- Divide both sides by 6

∴ x = 3

* The value of x is 3

water molecules will move from the side of higher water concentration to the side of lower concentration until both solutions are isotonic. at this point, equilibrium is reached. in an isotonic solution, the flow of water in and out of the cell is happening at the same rate.

hope this < 3

the correct answer is 828.07