3
$\begingroup$

In the diagram 4 circles of equal radius stand in a row in such a way that each circle touches the next one. $P$ is a point on the circumference of the first circle. The center of the fourth circle is point $Q$. The line $PQ$ goes through the centers of all four circles. $PC$ is a tangent on the fourth circle such that it intersects the second circle at points $A$ and $B$. Radius of the circles is $7$ and the length of $AB$ is $a\sqrt b$.
Find the length.

enter image description here

$\endgroup$
2
$\begingroup$

enter image description here

Find x by similar triangles.

Find y by Pythagoras theorem.

$\endgroup$
  • $\begingroup$ Oh... I missed that $\endgroup$ – Symon Saroar Dec 18 '15 at 13:29
2
$\begingroup$

Let $\angle CPQ=\alpha$. We have : $\sin \alpha=\dfrac{CQ}{QP}=\dfrac{7}{7^2}= \dfrac{1}{7}$.

The distance from PC tho the center of the second circle is $d=3\cdot 7 \cdot \sin \alpha=3$ so the chord $AB$ is $AB=2\sqrt{7^2-3^2}=4\sqrt{10}$

$\endgroup$

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.