1
$\begingroup$

I have a really simple-looking question, but I have no clue how I can go about solving it?

The question is

Find the exact value of $x$ in the following diagram:

diagram

Sorry for the silly/easy question, but I'm quite stuck! To use the cosine or sine rule, I'd need the angle opposite $x$, but I can't find that, cause I don't have anything else to help it along.

Thank You!

$\endgroup$
3
$\begingroup$

Call the top point $A$, the point on the bottom left $B$, and the point on the bottom right $C$. Draw the altitude from $A$ to $BC$, and call the foot of the altitude $D$.

Now $\triangle ABD$ has angles $30^\circ$, $60^\circ$, and $90^\circ$ respectively. Therefore $BD$ has length $3$ and $AD$ has length $3\sqrt3$. Furthermore, by the Pythagorean theorem, $DC$ has length $\sqrt{7^2 - (3\sqrt3)^2} = \sqrt{22}$.

Therefore $x = BC = BD + DC = 3 + \sqrt{22}$.

(Forgive my shoddy length notation.)

$\endgroup$
3
$\begingroup$

Use the cosine rule with respect to the 60 degree angle. Then you get an equation involving $x$ as a variable, Then you solve the equation for $x$.

$\endgroup$
  • $\begingroup$ Good hint. Better than the one I was typing up. $\endgroup$ – Ross Millikan Aug 18 '12 at 4:26
3
$\begingroup$

Actually,you can use the Law of Cosine as follows:

$$ \rm{Cos}\,60^{\circ}=\frac{6^2+x^2-7^2}{2\times6\times x}, $$

Then you can easily find out that $x=3+\sqrt{22}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.