Question

Please help?

Answer 1

36

Explanation:

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`

Answer 2

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)`