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Side Length Settlement 2 (Posted on 2015-04-11) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
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Solution 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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