1
$\begingroup$

The diagram shows two circles with centres $A$ and $B$ which touch at $C$ and have radius $r$.

The points $D$ and $E$, one on each circle, are such that $DE$ is parallel to the line $ACB$. Each of the angles $DAC$ and $EBC$ is $\theta$ radians where $0<\theta<\pi$.

Express the length of $DE$ in terms of $r$ and $\theta$.

I tried constructing a triangle drawing a line from $D$ to $C$. Used law of cosines to find $DC$ and compared the similarity of triangles $ADC$ and $DCE$, to find $DE$ but got $DE = 2r^2 - 2r^2cos\theta$ which is wrong. Hints, suggestions needed.

P.S. The length of DE at the back of book is given as $DE = 2r - 2r\cos\theta$.

$\endgroup$
3
$\begingroup$

Hint: drop perpendiculars from $D$ and $E$ onto $AB$. This yields a rectangle and two congruent triangles, all of which can be easily solved.

$\endgroup$
1
  • $\begingroup$ Thank you. This solved it. $\endgroup$ – AgentRock Jul 17 '17 at 14:17

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.