Are there any relatively easy problems that make use of the Intersecting Chord Theorem? And same question for Intersecting Secants Theorem

Question

I'm talking about the theorems as laid out here:

https://www.mathopenref.com/chordsintersecting.html

https://www.mathopenref.com/secantsintersecting.html

They require you to know these for iGCSE maths.

Are there any problems where you can make use of these theorems, in order to practice for the iGCSE exams? Obviously the problems can't be too difficult because it's only GCSE level. Maybe I should ask this on Math Education SE instead?

My motivation for this is: I'm a maths tutor and haven't found many problems my GCSE students can practice applying these theorems on...

Is it simply the case they need to be able to find the 4th length if they are given 3 out of the 4 lengths? That seems likely actually, at GCSE level...

Answer 1

In the following, $A,B,C,D$ are concyclic, and $P$ is the intersection point of $AD$ with $BC$.

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Question 1 : Find $PC$ when $AP=4,AB=5,PD=2,\angle{APB}=90^\circ$.

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Question 2 : Find $PA$ when $AB=8,BC=7,CA=6,BD=1,\angle{BAP}=\angle{CAP}$.

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Question 3 : Find $PO$ when $PA=2,AD=4,OC=3$ where $O$ is the center of the circle.

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Question 4 : Find $PA$ when $AB=3,AD=4,DC=6,BC=2$.

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Question 5 : Find $PB$ when $PC=12,CD=5,PA=4,\angle{PCD}=90^\circ$.

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