can someone explain the conclusion underlined in RED (see attached image) ?
Consider triangle PQS -
given angle S = 90
if angle P = x
then angle PQS = 90 -x
and triangle QSR -
given angle S = 90
if angle R = y
then angle SQR = 90 -y
Nothing in the data leads to x = y OR x = 90 -y OR y = 90 -x
i.e. the inner triangles share only one common angle (the right angle) - then how are triangles PQS and QSR similar ? (or in other words, how are all three triangles similar ?)
In fact, the problem goes on to say that the ratio of the areas of triangles PQS and RQS are in the ratio of the square of the respective side lengths. That seems incorrect.
Consider,
area triangle PQS = 1/2 (PS)(QS)
area triangle QSR = 1/2 (SR)(QS)
so the areas are in the ratio = PS/SR, which is the ratio of respective side lengths, and not square of the respective side lengths. Should'nt the answer be 4/3 ?
