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.

Answer:a) The length of the mid-segment  is 6.25 cmb) The length of AT = 33 unitsc) The value of x is 3Step-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 cmb)∵ 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 unitsc)- 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