Segment Products and Power of a Point: Practice

Secants have orders $P,A,B$ and $P,C,D$. The lengths $PA,AB,PC,PD$ are $x,x+6,4,27$, respectively. Here $x$ itself is a length. Find $x$.

A tangent from $P$ has length $x+3$. A secant has external length $x$ and inside chord length $15$. Find the valid $x$.

Chords have orders $A,P,B$ and $C,P,D$. Their lengths are $x+6$, $6-x$, $5$, and $7$, respectively. Here $x$ is a parameter. Which gives every valid $x$?

A circle has radius $13$ and $OP$ is $5$. A chord through $P$ has $PA$ of $9$. Find $PB$.

An external point has signed power $91$ and center distance $10$. Find the diameter of the circle.

Chord $AB$ has length $14$ and contains interior $P$. Write $PA=x$, so $PB$ is $14-x$. Another chord through $P$ has parts $4$ and $10$. Which lists every possible $PA$?

Secants have orders $P,A,B$ and $P,C,D$. The lengths $PA,PB,PC,CD$ are $2,18,3,9$. A tangent from $P$ touches at $T$. Which pair gives $AB$ and $PT$?

From $\triangle PTA\sim\triangle PBT$, which proportion directly gives the tangent-secant product for a general configuration?

A point $P$ is $5$ units from the center of a circle and has signed power $-50$. Find the radius.

A diameter lies on a secant ray in order $P,A,O,B$, where $O$ is the center. The lengths $PA,PB$ are $7,19$. Find $OP$.

A secant from external $P$ has near and far lengths $x+2$ and $x+9$. Its tangent segment has length $2\sqrt{15}$. Here $x$ is a parameter. Find the valid $x$.

A circle has radius $\sqrt{85}$. An interior point $P$ has center distance $6$. An exterior point $Q$ has center distance $11$. Which pair gives the positive chord-part product at $P$ and a tangent length from $Q$?