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

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step-by-step explanation: