perplexus.info :: Geometry : Acutely Seeking Perimeter

Acutely Seeking Perimeter (Posted on 2007-09-28)

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.

Submitted by K Sengupta

Rating: 4.0000 (1 votes)

Solution: (Hide)

We have UK = 3 = KR. Accordingly, K is the midpoint of UR and T is the midpoint of PR.

So KS is parallel to PQ, and hence S is the midpoint of QR. Therefore, PS corresponds to a median as well as an altitude, and accordingly PQ = PR. Also, PS bisects the angle QPR.

Now, Angle UPJ = Angle JSK; Angle PJU = Angle SJK, and accordingly the triangles PUJ and SKJ are similar.

Therefore, PJ/JS = UJ/JK = 2/1

Accordingly, J corresponds to the centroid of the triangle PQR. Since RU bisects the angle PRQ and it passes through J, it follows that RU is also a median. Consequently, RP = RQ

Thus, the triangle PQR is equilateral.

CF = Altitude of the triangle = 6, so that:
Length of each side of the triangle = 4√(3), giving:

The required perimeter of the triangle = 12√(3) = 20.78461 (correct to five places of decimals).

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Some helpful lines!	Chesca Ciprian	2007-09-28 14:49:13
Solution	Charlie	2007-09-28 11:05:16

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