PQR is an acute angled triangle.

It is known that:

• S, T and U are points located respectively on the sides QR, RP and PQ.
• PS is perpendicular to QR.
• RU is the internal bisector of the angle PRQ.
• RU intersects PS and ST respectively at the points J and K.
• PT = TR.
• UJ=2, JK = 1 and KR = 3

Determine the perimeter of the triangle PQR.

Let be TA perpendicular to QR

Let be UB perpendicular to QR and TA intersects UR in L

Because TA is is paralel with PS and T is in the middle of PR then A is in the middle of SR and L is in the middle of JR.

But JR is 4 so  LR=LJ=2. So JK=KL=1 and the triangles TKL and JKS are the same.

So TK=KS and KR is both bisector and median in the TRS triangle.

So the isoscel triangle TRS is in fact echilateral.

So R ungle is 60º.

Easy after this can find PR = 4*sqrt(3)and SR=2*sqrt(3).

To find PQ and QS we calculate first UB from the trinagle UBR.

So UB=3 and because PS=6 then UR is both bisector and median in triangle QRP. After this we find that QS=2*sqrt(3) and QP=4*sqrt(3). So PR=RQ=QP=4*sqrt(3) and the answer is 12*sqrt(3)!

 

Edited on September 28, 2007, 3:40 pm

Posted by Chesca Ciprian on 2007-09-28 14:49:13