perplexus.info :: Shapes : Side Length Settlement 2

Side Length Settlement 2 (Posted on 2015-04-11)

Points S and T are inside an equilatral triangle PQR such that:
ST = 1, SP = TP =√7, and:
SQ = TR = 2.

Find the length of PQ.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

Comment 1 of 1

Let PN be an altitude of triangle PQR, crossing ST at M.
Using PN as an axis of symmetry, /PSM = 90 deg. and
Pythagoras gives:
            PM = sqrt(7 – ¼) = 3*sqrt(3)/2

and       PN = a*sqrt(3)/2            where a = |PQ|

In triangle TRA (where TMNA is a rectangle):

            TA² + AR² = TR²

(a*Sqrt(3)/2 – 3*sqrt(3)/2)² + (a/2 -1/2)² = 2²

            3(a – 3)² + (a – 1)² = 16

            a² – 5a + 3 = 0

|PQ| = a = (1/2)(5 + sqrt(13)) = 4.302…

Posted by Harry on 2015-04-13 18:36:05

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