# Math Help - geometry finding length

1. ## geometry finding length

how do you find segment AD and segment DH?

i was thinking of the trapezoid and separating it into 3 different shapes; a rectangle, a triangle and a triangle.

2. Hello, algebra2!

Code:
          A 2 H
*---*
2/ |     \
B *---*P      * G
2|   |       |
|   |       |
C *---*Q      * F
2\ |     /
*---*
D 2 E
We have a regular octagon with side 2.
Draw $BP \perp AD \text{ and }CQ \perp AD$
Draw $DH.$

We have isosceles right triangle $APB$ with hypotenuse 2.
. . Hence: . $AP = \sqrt{2}$
Similarly: . $QD = \sqrt{2}$

Hence: . $AD \;=\;AP + PQ + QD \;=\;\sqrt{2} + 2 + \sqrt{2}$

. . Therefore: . $\boxed{AD \;=\;2 + 2\sqrt{2}}$

In right triangle $H\!AD\!:\;\;DH^2 \:=\:AH^2 + AD^2 \;=\;2^2 + (2+2\sqrt{2})^2 \;=\;16 + 8\sqrt{2}$

. . Therefore: . $\boxed{DH \:=\:2\sqrt{4 + 2\sqrt{2}}}$