Question

In a circle a chord 10 inches long is parallel to a tangent and bisects the radius drawn to the point of tangency. How long is the radius?

Answer 1

Check the picture below.

keeping in mind that the point of tangency for a radius line and a tangent is alway a right-angle, since the "red" chord is parallel to the "green" tangent line outside, then the chord is cutting the "green" radius there in two equal halves at a right-angle, as you see in the picture.

we know the chord is 10 units long, so 5 + 5, since is perpendicularity with the radius will also cut the chord in two equal halves.

anyhow, all that said, we end up with triangle you see on the right-hand-side, and then we can just use the pythagorean theorem.

$\bf \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2 \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ -------\\ c=2a\\ a=a\\ b=5 \end{cases}\implies (2a)^2=a^2+5^2$

$\bf (2^2a^2)=a^2+25\implies 4a^2=a^2+25\implies 3a^2=25 \\\\\\ a^2=\cfrac{25}{3}\implies a=\sqrt{\cfrac{25}{3}}\implies a=\cfrac{\sqrt{25}}{\sqrt{3}}\implies a=\cfrac{5}{\sqrt{3}} \\\\\\ \textit{and then we can \underline{rationalize the denominator} } \\\\\\ a=\cfrac{5}{\sqrt{3}}\cdot \cfrac{\sqrt{3}}{\sqrt{3}}\implies a=\cfrac{5\sqrt{3}}{\sqrt{3^2}}\implies a=\cfrac{5\sqrt{3}}{3}$