Prove: If two circles are tangent internally at point P and chord PA
of the larger circle intersects the smaller circle at B, then the measure of arc PA is equal to the measure of arc PB.

Proved

Step-by-step explanation:

Given:

Two circles are tangent internally at point P and chord PA of the larger circle intersects the smaller circle at B.

Prove: measure of arc PA is equal to measure of arc PB

To better understand this, the diagrams have been attached.

From triangle PQB:

Line PQ  and line QB are radius of the smaller circle.

Reason:  their line are straight lines  drawn from the center of the circle to the circumference of the circle

Line PQ = Line QB

That is, side PQ = side QB

Since side PQ = side QB, it is an isosceles triangle.

Therefore < QPB = <PBQ

From triangle PRA

Line PR = line RA

Side PR = side RA

From triangle PRA:

Line PR  and line RA are radius of the bigger circle.

Reason:  their line are straight lines  drawn from the center of the circle to the circumference of the circle

Line PR = Line RA

Since side PR = side RA, it is an isosceles triangle.

Therefore < RPA = <PAR

Using similar triangles theorem:

Triangle PQB = Triangle PRA

<PBQ = <PAR

Since they are equal, the measure of the arc (angle subtended by the arc) are equal.

Therefore measure of arc PA = measure of arc PB

5y

step-by-step explanation:

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.