In a parallogram ABCD, P divides AB in the ratio 2 : 5 and Q divides DC in
the ratio 3 : 2. If AC and P Q intersect at R, find the ratios AR : RC and P R : RQ.
Answers
Answer:
ABCD is a parallelogram with two of its sides equal and other two sides parallel to each other.
In ∆APR and ∆CRQ, we have
∠PAR = ∠RCQ ….. [alternate angles]
∠RQC = ∠APR …… [alternate angles]
∠PRA = ∠QRC ….. [vertically opposite angles]
∴ By AAA criterion, we get
∆APR ≅ ∆CRQ
∴ AP/CQ = AR/RC = PR/RQ ….. (i) [corresponding sides of similar triangles are proportional to each other]
From the figure below, we can write
AP / AB = 2/(2+5) = 2/7
⇒ AP = (2/7) * AB …… (ii)
And,
PB / AB = 5 /7
⇒ PB = (5/7) * AB …… (iii)
Since ABCD is a parallelogram, therefore, AB = CD ….. (iv)
So, from the figure and (iv), we get
CQ = (2/5) * AB and DQ = (3/5) * AB ……. (v)
From (i), (ii) & (v), we get
AR/RC = PR/RQ = AP/CQ = [(2/7)*AB] / [(2/5)*AB]= 5/7
Thus, the ratio of AR:RC & PR:RQ is 5:7.
