Power of a Point Theorem/Introductory Problem 3

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Problem

(ARML) In a circle, chords $AB$ and $CD$ intersect at $R$. If $AR:BR = 1:4$ and $CR:DR = 4:9$, find the ratio $AB:CD.$

Solution

Letting $AR = x$ makes $BR = 4x$. Similarly, letting $CR = 4y$ makes $DR = 9y$. Thus $AB = AR + BR = x + 4x = 5x$ and $CD = CR + DR = 4y + 9y = 13y$. We therefore seek $\frac{AB}{CD} = \frac{5x}{13y}$.

From the Power of a Point Theorem, we have that

$x\cdot 4x = 4y\cdot 9y\Rightarrow \left(\frac xy\right)^2 = 9$,

which gives $\frac xy = \pm 3$, so we take $\frac xy = 3$.

Finally,

$\frac{5x}{13y}=\frac 5{13}\cdot \frac xy = \frac 5{13}\cdot 3 = \frac{15}{13}.$

Back to the Power of a Point Theorem.