Please help?

When a line is a perpendicular bisector of another line, then that means that it divides the line into 2 equal parts. It also means that the lines are at right angles to each other
For example, if AC is a perpendicular bisector of DE, then DB=BE

If we know that DB=BE, AC is at right angles to DE and AB is a common side of #triangle ABE# and #triangle ABD#, then we can assume that AD=AE.

Why? By using congruent triangles

In #triangle ABE# and #triangle ABD#,
1. AB is the common side
2. DB=BE (AC is a bisector of DE)
3. AC is at right angles to DE (AC is a perpendicular bisector of DE)
then #triangle ABE = triangle ABD# (SAS or two sides and one angle between the two sides are equal)

Therefore, AD=AE (same sides of proven congruent triangles are equal)

#3x-9=x+21#
#2x=30#
#x=15#

Since we want to find side AE which is equal to #x+21#, we can sub in #x=15# to find its value which is #15+21=36#

Since #bar(AC)# perpendicularly bisects #bar(DE)#,

• the lengths of #bar(DB)# and #bar(BE)# are the same (since #bar(DE)# is divided in two equal parts), and angle #/_ABE# is #90^@# (since #bar(AC)# intersects #bar(DE)# perpendicularly).
• triangle #DeltaDBA# is a horizontal reflection of triangle #DeltaABE# (since #bar(AC)# divides one triangle #DeltaDAE# into two identical halves).

It then follows from the triangles being a reflection of each other that the lengths of #bar(AD)# and #bar(AE)# are identical. Therefore:

#3x - 9 = x + 21#

#=> 2x - 9 = 21#

#=> 2x = 30#

#=> x = 15#

As a result, the length of #bar(AE)# is:

#color(blue)(L) = x + 21 = 15 + 21 = color(blue)(36)#