Question

Line X Y is a secant to the larger circle and is tangent to the smaller circle at point Q. The radius of the

smaller circle is 7 and the radius of the larger circle is 9. What is the length of chord XY?

Answer 1

`8√(x) 2 - 11.31\\`

Answer 2

Check the picture below.

so the chord we see in the picture in "green", has a length of XQ + QY, now, we can just use the pythagorean theorem to get the length of XQ, keeping in mind that the point of tangency at Q is always a right-angle, and thus that means that XQ = QY, so whatever the green chord is, is just really 2 * XQ.

`\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies √(c^2 - a^2)=b \qquad \begin{cases} c=\stackrel{hypotenuse}{9}\\ a=\stackrel{adjacent}{7}\\ b=\stackrel{opposite}{XQ}\\ \end{cases} \implies √(9^2 - 7^2)=XQ \\\\\\ √(32)=XQ\implies 4√(2)=XQ\hspace{5em} \underset{\textit{\LARGE XY}}{\stackrel{(2)4√(2)}{{\Large \begin{array}{llll} 8√(2)\approx 11.31 \end{array}}}}`