Draw a segment that connects the points A and P. Then, use the right triangle △ APG to find the radius of ⊙ P.
17
Practice makes perfect
Let's consider the given diagram.
We know that AC=FD=30, PG=x+5, and PJ=3x-1. Now let's connect the points A and P with a segment.
Since the endpoints of AP are the center and a point on the circle, it is a radius of ⊙ P. Now we can determine the radius of ⊙ P. Note that △ APG is a right triangle because PG⊥AC. Therefore, by the Pythagorean Theorem we have the following.
PG^2+AG^2= AP^2
Let's determine the values of PG and AG using the properties of chords. Then, we will use the equation above to find the radius AP.
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
Recall that AC=FD=30, so AC and FD are congruent chords. Therefore, these two chords are equidistant from the center P. The distance of AC from P is PG, and the distance of FD from P is PJ.
PG=PJ
Let's substitute x+5 for PG and 3x-1 for PJ and solve the equation for x.
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Now let's take a look at the diagram.
We can see that EB is the diameter of ⊙ P. Additionally, EB is perpendicular to AC.
EB ⊥ AC
Therefore, EB bisects AC, and AG=GC= 12AC. We know that AC= 30, so let's find AG.
Let's recall the equation that we wrote at the start using the Pythagorean Theorem.
PG^2+AG^2=AP^2
We already know that PG= 8 and AG= 15. We will substitute these values into the above equation and simplify.
