Two Red Lines in a Rectangle

Imagine a rectangle with sides $\overline{\text{AB}}$, $\overline{\text{BC}}$, $\overline{\text{CD}}$, and $\overline{\text{DA}}$. Side $\overline{\text{AB}}$ has a length of $4$ while $\overline{\text{BC}}$ has a length of $6$.

Put point $\text{E}$ on $\overline{\text{AD}}$ . Draw a line from $\text{E}$ to $\text{B}$ and draw another line from $\text{E}$ to $\text{C}$ .

If $\angle\text{BEC}$ has a measurement of $65°$ , what is the length of line segments $\overline{\text{EB}}$ and $\overline{\text{EC}}$ ?

$Rectangle $\text{ABCD}$$ Rectangle $\text{ABCD}$

How can I solve for the lengths of the red lines? A step-by-step solution will be very much appreciated.

Here's the link of an image of described shape: http://imgur.com/gallery/x6kGdAq

#Geometry

Easy Math Editor

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Comments

Here's a hint: let $\overline{DE}$ be $x$ . Then, $\overline{AE} = 6-x$ . We can use the Pythagorean Theorem to express $\overline{EB}$ and $\overline{EC}$ in terms of $x$ . Then, you can use Cosine Law on $\triangle EBC$ and solve for $x$ . Plug the result back into $\overline{EB}$ and $\overline{EC}$ , and the problem is solved!

Elijah L - 1 year, 1 month ago

$\overline{AE}$ is equal to $6-x$ ? But $\overline{AD}$ is $6$ and you said let it be $x$ .

Kaizen Cyrus - 1 year ago

Whoops, my mistake. I meant let $\overline{DE}$ be $x$ . Everything else stays the same.

Elijah L - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$