1. ## Ptolemy's Theorem and Cyclic Quadrilaterals

In need of some help for 3 problems that I can't seem to understand. I would greatly appreciate some help.

Point P on side AB of a right Triangle, ABC, is placed so that BP = PA = 2. Point Q is on hypotenuse AC such that PQ is perpendicular to AC. If CB = 3 what is the measure of BQ?

What is the area of quadrilateral CBPQ?

As P is translated from B to A along BA, find the range of values of BQ, where PQ remains perpendicular to CA.

I don't understand the part where you translate P along BA at all. I can't seem to even draw the diagram correctly for the quadrilateral. I am also having a hard time drawing the diagram to even solve the first part, I just know the pyhgreoen theorem might help out. Thank you so much for the help.

2. i think the diagram comes likes this

3. Oh thank you so much for the diagram ! This helps a lot but I'm still wondering how I can find the measure of BQ when I connect Point B to Q. When I do that is the angle PBQ a right angle ?

4. i doubt it.

i think you were able to find the length of AQ.
If so, since you know AB and $\angle PAQ$ you can use cosine equation to find BQ

5. Originally Posted by zzz123
Oh thank you so much for the diagram ! This helps a lot but I'm still wondering how I can find the measure of BQ when I connect Point B to Q. When I do that is the angle PBQ a right angle ?
1. You are dealing with similar right triangles:

$\Delta(AQP) \sim \Delta(ABC)$

and

$\Delta(QFC) \sim \Delta(ABC)$

2. The distance BQ is the hypotenuse of the right triangle $\Delta(BFQ)$

6. I'm not sure I follow. How do I find length of AQ ? Im given AB but unsure of what Angle PAQ.

Oh seeing that they are similar helped a lot ! But now im unsure on how to find the length of QF and FC.

Thanks again for both your help.