# Math Help - perpendicular segment measure

1. ## perpendicular segment measure

Let ABCD be a square with each side of length 2cm. Let E be the midpoint of AB. How long is the perpendicular segment C to DE?

can anyone help me on this please

2. ## Re: perpendicular segment measure

make a sketch ... use the fact that the sides of similar triangles are proportional.

3. ## Re: perpendicular segment measure

is this correct 2* sqrt(5) / 5

4. ## Re: perpendicular segment measure

JHello, rcs!

Let ABCD be a square with each side of length 2cm.
Let E be the midpoint of AB.
How long is the perpendicular segment from C to DE?

Code:
            1     E
A o - - - - - o - - - - - o B
|        θ /            |
|         /             |
|      F /              |
|    _  o               |
|   √5 /  *             |
2 |     /     *           |
|    /        *         |
|   /           *       |
|  /              *     |
| /                 *   |
|/ θ                  * |
D o - - - - - - - - - - - o C
2
In right triangle $EAD$, the hypotenuse is: $DE \,=\,\sqrt{1^2+2^2} \,=\,\sqrt{5}$

Right triangles $EAD$ and $DFC$ have angle $\theta$.
. . Hence: . $\Delta EAD \sim \Delta DFC$

$\begin{array}{cccccccc}\text{In }\Delta DFC:& \sin\theta &=& \dfrac{CF}{2} \\ \\[-3mm] \text{In }\Delta EAD: & \sin\theta &=& \dfrac{2}{\sqrt{5}} \end{array}$

Therefore: . $\frac{CF}{2} \:=\:\frac{2}{\sqrt{5}} \quad\Rightarrow\quad CF \:=\:\frac{4}{\sqrt{5}} \;=\;\frac{4\sqrt{5}}{5}$