Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


In the given circle |AB| = 10 Units and AB || CD

AB is the diameter of the circle.

What is the length of the chord ED?

share|cite|improve this question
Is $AB$ supposed to be a diameter of the circle? – Robert Israel Sep 14 '12 at 7:37
Yes AB is the diameter of the circle. – Sumit Bhowmick Sep 14 '12 at 7:38
I would be intrigued to know if it is possible without relying on trigonometry for the final calculation. – Arthur Sep 14 '12 at 10:29
up vote 3 down vote accepted

By inscribed angle of a cirle, $\angle CBE = \angle CDE = 15^\circ$. That means $\angle ABE = 25^\circ$. Since $AB$ is a diameter, $\angle AEB$ is right, and thus we conclude that $\angle BAE = 90^\circ - 25^\circ = 65^\circ$.

Since $\angle BAD = 10^\circ$ by symmetry, we have that $\angle DAE = 65^\circ - 10^\circ = 55^\circ$. If we let $O$ be the center of the circle, we know then that $\angle DOE = 2\cdot\angle DAE = 110^\circ$. Your unknown side is now the last side of an isosceles triangle with two sides $5$ long, since they are radii in the circle, and with angles $110^\circ$, $35^\circ$ and $35^\circ$. That should be easer to calculate.

share|cite|improve this answer

Common case $\angle ABC = \alpha, \angle CDE = \beta$: enter image description here

Let $DF || BE \Rightarrow |ED|=|BF|$.

$\angle CBE = \angle EDC = \beta$ - same chord. $\angle EBF = \angle BFD= \angle DCB= \angle ABC =\alpha$.

As $\angle AFB=\frac{\pi}{2} \Rightarrow |FB|=|AB|\cdot \textbf{cos} \angle ABF$.

So $|ED|=|AB|\cdot \textbf{cos} (2\alpha+\beta)$.

share|cite|improve this answer
Sorry to bother but what tool do you use to draw images like the one you posted? – Gabber Sep 14 '12 at 17:26
GeoGebra. – Mike Sep 14 '12 at 17:51
Many many thanks – Gabber Sep 14 '12 at 17:53
@Mike: Manu many thanks – Sumit Bhowmick Sep 28 '12 at 4:57

enter image description here

O is center of circle. and You must know that total angle of circle is 360 degree.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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