[Math] Find length of chord between two tangents and distance of chord from origin

circlescongruences-geometrygeometrytangent line

I am trying to solve 2 questions

Is there a way to find an equation for the length of a chord between two tangents in terms of the radius and distance of that chord to the external point?

Is there a way to find an equation for the distance between that chord and the center of the circle in the same terms (radius and distance of chord to external point)?

Please show working/proof

Here is a graphical representation

Lets define some terms:

$w = AB$

$t = OI$

$h = IP$ ie height of $ΔABP$

$r = OB$ ie radius

$AP$ and $BP$ are tangents to the circle

I am wanting to find:

  • An equation for $w$ in terms of $h$ and $r$ ie "$w = …$"

  • An equation for $t$ in terms of $h$ and $r$ ie "$t = …$"

Is this actually possible??

I have had a go and concluded the following:

$ΔBOP|||ΔAOP$ (radius, shared side + 90deg) (congruent)

$ΔBOP☰ΔIBP$ (shared angle, 3 angles) (similar)

$ΔIBP☰ΔIOB$ ($∠IOB=∠IBP$ [Angle of chord to tangent is half angle of sector created by chord], 3 angles) (similar)

Hence $ΔBOP☰ΔIBP☰ΔIOB$

But I don't no where to go from there

Thanks

UPDATE
So using the similar triangles $\Delta IOB \sim \Delta IBP$ we can state:
$$
\frac{w}{t}=\frac{h}{w}, \therefore w^2 = th
$$

We can also use Pythagoras Theorem on $\Delta IOB$
$$
\frac{w}{t}=\frac{h}{w}, \therefore w^2 = th
$$

Best Answer

The hint.

Since $BL$ is an altitude of $\Delta OBP$ and $\measuredangle OBP=90^{\circ},$ we obtain $$w^2=th$$ and $$w^2=h\sqrt{r^2-w^2}.$$ Now, solve the last equation.

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