3560.11 – Intersecting Chords

Two chords intersect in a circle. The measures of the segments of one chord are $2^x$ and $5^x$ . The measure of the segments of the second chord are $4^{x-1}$ and $5^{x-1}$ . Consult the figure for the exact relation of these segments.

Then find the numerical value of $2^x$ .

Solution

By similar triangles, $2^x \cdot 5^x = 4^{x-1} \cdot 5^{x-1}$ . Thereafter, a little manipulation gives,

$\begin{aligned}2^x \cdot 5^x &= 2^{2x-2} \cdot 5^{x-1} \\\dfrac{2^x}{2^{2x-2}} &= \dfrac{5^{x-1}}{5^x} \\2^{-x+2} &= 5^{-1} \\\dfrac{2^2}{2^x} &= \dfrac{1}{5} \\20 &= 2^x\end{aligned}$

The answer is $20$ .