Question 712·Hard·Math

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A circle has center $P$ , and points $M$ and $N$ lie on the circle. Line segments $MX$ and $NX$ are tangent to the circle at $M$ and $N$ , respectively. If the radius of the circle is $315$ centimeters and the perimeter of quadrilateral $PMXN$ is $1{,}470$ centimeters, what is the distance, in centimeters, between points $P$ and $X$ ?

Two tangent segments from the same outside point are equal, so the perimeter can reveal their shared length. A radius to a tangent point then makes a right triangle whose hypotenuse is the center-to-outside-point distance. Solve the perimeter equation with a Desmos regression, then look for a scaled $3$ - $4$ - $5$ triangle. Don’t stop at the tangent length when the question asks for a different segment.

Hints

Hint 1
A radius connects the center to a point on the circle, so $PM$ and $PN$ have the same length. The two tangent segments from $X$ are equal too. Which four sides make up the perimeter?
Hint 2
A perimeter adds the sides around a figure, not its diagonals. Call each tangent length $x_1$ , then add the two radii and two tangents to match the given total.
Hint 3
At a point of tangency, the radius and tangent form a right angle. In triangle $PMX$ , which side is across from that angle? Use the two shorter sides to find its length.

Step-by-step

Find the tangent, then the hypotenuse
Step 1
Identify the two radii
A radius runs from the center to a point on the circle. Since $M$ and $N$ lie on the circle, both $PM$ and $PN$ measure $315$ centimeters.
Step 2
Match the tangent lengths
Two tangent segments from the same outside point are equal. Both tangents start at $X$ , so $MX=NX$ . Call their common length $x_1$ centimeters.
Step 3
Write the perimeter equation
A perimeter adds the sides around the quadrilateral: $PM$ , $MX$ , $XN$ , and $NP$ . The segment $PX$ is a diagonal, so it isn’t included. Substitute the four side lengths:
$315+x_1+x_1+315=1{,}470.$
Step 4
Find each tangent length
Type $315+x_1+x_1+315\sim1470$ in Desmos. The $\sim$ tells Desmos to find the value of $x_1$ that fits the equation. Under PARAMETERS, it shows $x_1=420$ , so each tangent is $420$ centimeters. That isn’t $PX$ ; it’s $MX$ or $NX$ .
Step 5
Locate the right triangle
A radius to a tangent point is perpendicular to the tangent. So $PM$ and $MX$ meet at a right angle at $M$ . Triangle $PMX$ has legs $315$ and $420$ ; $PX$ is the hypotenuse, the side across from that right angle.
Step 6
Scale a familiar right triangle
The legs $315=3(105)$ and $420=4(105)$ match the $3$ - $4$ - $5$ Pythagorean triple, a set of right-triangle side lengths. Scale its hypotenuse by the same amount: $PX=5(105)=525$ . The distance from $P$ to $X$ is $525$ centimeters. Choice C.

Question 712·Hard·Math

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