3560.11 – Intersecting Chords

Two chords intersect in a circle. The measures of the segments of one chord are 2x2^x and 5x5^x. The measure of the segments of the second chord are 4x−14^{x-1} and 5x−15^{x-1}. Consult the figure for the exact relation of these segments.

Then find the numerical value of 2x2^x.

3560_11_fded36e70a.png

Solution

By similar triangles, 2x⋅5x=4x−1⋅5x−12^x \cdot 5^x = 4^{x-1} \cdot 5^{x-1}. Thereafter, a little manipulation gives,

2x⋅5x=22x−2⋅5x−1,2x22x−2=5x−15x,2−x+2=5−1,222x=15,20=2x. \begin{aligned} 2^x \cdot 5^x &=& 2^{2x-2} \cdot 5^{x-1}, \\ \frac{2^x}{2^{2x-2}} &=& \frac{5^{x-1}}{5^x}, \\ 2^{-x+2} &=& 5^{-1}, \\ \frac{2^2}{2^x} &=& \frac{1}{5}, \\ 20 &=& 2^x. \end{aligned}

The answer is 2020.