Question

15 multiple choice 1 point in the diagram, \\( \overrightarrow { p x } \\) is a tangent and \\( \overline { o x } \\) is a radius. if the length of \\( \overline { o x } \\) is 5 and \\( o p = 13 \\), how long is \\( \overline { p y } \\)? 12 \\( \sqrt { 145 } \\) \\( \sqrt { 194 } \\) 8

Explanation:

Step1: Use the property of tangent and radius

A tangent to a circle is perpendicular to the radius at the point of tangency. So, $\angle{OXP} = 90^{\circ}$. In right - triangle $OXP$, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$ (where $c = OP$, $a=OX$, $b = PX$).

Step2: Calculate the length of $PX$

Given $OX = 5$ and $OP = 13$. Substitute into the Pythagorean theorem: $PX=\sqrt{OP^{2}-OX^{2}}=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$.

Step3: Use the property of tangents from a common external point

Tangents from a common external point to a circle are equal in length. Since $PX$ and $PY$ are tangents from point $P$ to the circle, $PX = PY$.

Answer:

A. 12