A. find the length of the mid segment 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 for 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) Find the length of the mid segment of an equilateral triangle with side lengths of 12.5 cm

Solution:

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 always parallel to the third side, and the length of the mid segment is half the length of the third side.

This is an equilateral triangle with side lengths 12.5cm

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

The length of the mid segment = 1/2 * 12.5

The length of the mid segment = 6.25 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

Solution:

UT is the perpendicular bisector of AB

T lies on AB

AT = BT

3x+6 = 42-x

Combine the like terms:

3x+x=42-6

4x= 36

Divide both sides by 4

4x/4 = 36/4

x= 9

Now plug the value of x in AT= 3x+6.

=3(9)+6

=27+6

=33

The length of AT = 33 units.

C) Given angle ABC has angle bisector for BD, where AB = CB, find the value of x if AD = 5x + 10 and DC = 28 - x.

Solution:

In Δ ABC

AB = BC

Δ ABC is an isosceles triangle

BD bisects angle ABC

AC is the opposite side of the vertex B

BD bisects the side AC at D

AD=CD

AD=5x+10

CD= 28-x

Equate both the equations:

5x+10 = 28-x

Combine the like terms:

5x+x=28-10

6x=18

Divide both sides by 6

6x/6 = 18/6

x= 3

Thus the value of x is 3