# [Math] Find the value of x in the diagram.

circlesgeometry

Consider a circle whose radius is $5 m$. Let $AB$ be a chord in the circle of lenth $6 m$. And the two tangents at the points $A$ and $B$ are intersect at the point $P$.

In the following diagram, $CP=x$ and $AB=6$ and $OD=5$

How to find the value of $x$?

It is easy to see, that $OP$ and $AB$ are perpendicual and intersects $AB$ in two equal parts, ie. $|AD|=|BD|=3$.
After Pythagoras: $$|AD|^2 + |OD|^2 = |OA|^2$$ $$3^2 + |OD|^2 = 5^2$$ we have then $|OD|=4$
Let $\alpha = ∠DOA$ and $|OP|=y$ Then $$\cos \alpha = \frac{|OD|}{|OA|} = \frac{|OA|}{|OP|}$$ $$\frac{5}{y}= \frac{4}{5}$$
Then $y = \frac{25}{4}$, and $x=y-5=\frac{5}{4}$