Use the diagram. Line FD is the perpendicular bisector of segment AC. Find the indicated measure.

With the assumption that the line FD passes through the point B, we have;

y = 7

x = 6

Step-by-step explanation:

The given parameters are;

Line FD bisects segment AC

Therefore, segment AE = segment EC By definition of line AC which is bisected by the line FD

Segment ED ≅ Segment ED by reflexive property

∠CED = ∠AED = 90° (Angles formed by a perpendicular bisector (FD) to a line (AC))

Therefore;

ΔCDE ≅ ΔADE by Side-Angle-Side (SAS) rule of congruency

From which we have;

Segment CD ≅ Segment AD Congruent Parts of Congruent Triangles are Congruent (CPCTC)

Segment CD = Segment AD Definition of congruency

∴ 12·y - 8 = 8·y + 20 by substitution property

12·y - 8·y = 20 + 8

4·y = 28

y = 28/4 = 7

y = 7

From segment AE = segment EC, we have;

2·x + 4·y = 2·x + 4·y by substitution property

2·x + 4×7 = 2·x + 4×7 by substitution property

Segment AE = 2·x + 28 = Segment EC

Segment AC = Segment AE + Segment EC by definition of segment (AC) bisected by a line (FD)

∴ Segment AC = 2·x + 28 + 2·x + 28 = 4·x + 56 by substitution property

Segment CD = 8·y + 20 = 8 × 7 + 20 = 56 + 20 = 76

Segment CD = 76

The sides of the ΔABC are;

Segment BC = 6·x + 18

Segment BA = 8·x + 6

Segment AC = 4·x + 56

With the assumption that the Line FD passes through the point B, we have;

Segment BC = Segment AB by congruent triangles ΔABE ≅CBE based on Side-Angle-Side (SAS) rule of congruency

Therefore;

6·x + 18 = 8·x + 6

18 - 6= 8·x - 6·x = 2·x

2·x = 18 - 6 = 12

x = 12/2 = 6

x = 6