In △abc, point p∈ab so that ap: bp=1: 2. point r ∈ pb that pr: br=1: 2. point q∈bc so angle bqr = angle bcp. find ratio of pqc to abc.

Ratio of PQC to ABC is 2:9.

Step-by-step explanation:

In ΔBRQ and ΔBPC

∠BQR = ∠BCP (given) and ∠B is common for both triangles, so from AAA similarity ΔBRQ and ΔBPC are similar.

⇒ BQ : QC = BR : RP = 2 : 1    →(1)

now draw perpendiculars from points A and P to BC line segment. Call the projected points as A' and P'.

It is clear that lines AA' and PP' are parallel. So in ΔBPP' and ΔBAA' we have AAA similarity with common angle at B.

⇒PP' : AA' = BP : BA = 2 : 3    →(2) (∵ AP : PB = 1 : 2)

area of ΔPQC = 0.5×PP'×QC

area of ΔABC = 0.5×AA'×BC

area of ΔPQC : area of ΔABC = (0.5×PP'×QC)/(0.5×AA'×BC)

                                               =(PP'/AA')×(QC/BC)

                                               =(2/3)×(1/3) (∵ from (1) and (2))

                                               =2/9.

∴ Ratio of PQC to ABC is 2:9

this is the formula: v = the value of investment at the end of the time period

p = the principal amount (the initial amount invested)

r = the annual interest rate

n = the annual frequency of compounding (how many times a year interest is added)

t = the number of years the money is invested

^ means raise to the power of