Question

10. DG, FG, and EG are perpendicular bisectors

Answer 1

AD = 5, x = 7, BC = 66

Explanation:

a) We know that DG is a perpendicular bisector, so it should divide AB into two equal parts, each equal to half the length of the total line segment.
So, AD = AB/2
= 10/2
= 5

b) Now, BE and EC are part of the same segment, as they share a common point E. Also, at E we have perpendicular bisector EG.
Again, the two parts BE and EC should be equal.
Given, BE = 3x + 12 and EC = 6x - 9
3x + 12 = 6x - 9
3x = 21
x = 7

c) Now, BC would be twice the length of one of the segments divided by the perpendicular bisector. So, BC = 2BE
= 2 * (3x + 12)
= 2 (3*7 + 12)
= 2 * 33
= 66

Answer 2

Explanation:

A) To find the length of AD, we first find the length of the CD.

CD = (BE + EC)/2 = (3x + 12 + 6x - 9)/2 = 9x + 3

AD = CD = 9x + 3

B) To find x, we use the fact that DG is the perpendicular bisector of EC and BG. This means that EG = BG = x.

EG = BG = x

EC = 2EG = 2x

Substituting in EC = 6x - 9, we get:

2x = 6x - 9

4x = 9

x = 9/4 = 2.25

C) To find BC, we use the fact that BC = AB - 2BE

BC = AB - 2BE

BC = 10 - 2(3x + 12)

Substituting in x = 2.25, we get:

BC = 10 - 2(3 * 2.25 + 12)

BC = 10 - 2 * 21

BC = 10 - 42

BC = -32