As the picture looks, is not a radius of circle O.
In triangle ACP we have .
Then applying cathetus' theorem we have
In circle O and circle P, AP is equal to 10 and AC is equal to 8. Line segment CB is also perpendicular to line segment AP. Find the measure of DP.
Here's what I have so far : Since all radii of a circle are congruent, triangle CBA is an isosceles triangle. And since base angles of an isosceles triangle are congruent, angle C and angle A are both equal to 45. After there, I guess I could use trig to figure out the measures of CB and go from there, but I think I'm going in the wrong direction.
Can anybody help me?
If you are allowed to use trigonometry, here's a start.
Connect CP. Then triangle ACP is a right triangle with angle ACP the right angle. So angle BAC (call this ) can be found by:
Triangle ABC is also a right triangle, so we know that
Can you take it from here?
(Another hint: angle PCB is also .)
red_dog: Thanks for the help, but I haven't learned cathetus' theorem, so I can't use that.
topsquark: If PCB is also equal to a theta, then does that mean I have to redo the work that you did for BC to find the measure of BP? Then would I draw axillary line DO and use the Pythagorean Theorem? But to be able to use that, I would have to prove that triangle BPD is a right triangle.
Sorry, but could you help me some more?
(I'm not sure if you can though since, you appeared to have signed off)
What's the length of the new line CP? (Look at triangle ACP.) It's 6. We know the sine of angle CBP and you know the hypotenuse (CP) of the right triangle BCP, so you can find the leg BP. And since BP = PD...
In circle and circle and
Find the measure of
Let , then
Since is inscribed in a semicircle,
. . it is a right triangle with .
We have: .
In right triangle .
Since is the altitude to the hypotenuse of a right triangle,
. . it is the mean proportional between the two segments of the hypotenuse.
Hence, we have: .
And we have: . .
Substitute  into : .