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Triangle Area Ascertainment (Posted on 2015-12-26) Difficulty: 3 of 5
PS, QT and RU are medians of triangle PQR. PS lies along the line y = x+3, QT lies along the line y = 2x+4.
The length of PQ is 60 and ∠PRQ = 90o.

Determine the area of triangle PQR.

No Solution Yet Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Solution Solution | Comment 1 of 2
Having used the coordinates of P, Q, S, T as variables, solved
simultaneous quadratics and struggled with decimals for some
time, I got the answer 399.999 and guessed that something
was afoot.
Here’s a shorter way (much fun – thanks KS).

Denote the lengths |RQ| and |RP| by a and b respectively.

By Pythagoras,               a2 + b2 = 602                 (1)

Let G be the intersection of the medians, so that GQ and GS
have gradients 2 and 1 respectively. Using these as tangent
ratios:  
            tan/QGS = (2 – 1)/(1 + 2*1) = 1/3          (2)

From triangle QGS:   /GSR = /QGS + /SQG  (exterior angle…)

Taking tangents of both sides..

    tan/GSR = (tan/QGS + tan/SQG)/(1 - tan/QGS*tan/SQG)

Using (2) and length ratios in triangles PSR and RQT,

we get         2b/a  = (1/3 + b/(2a))/(1 – (b/(6a))

which simplifies (eventually) to    ab = 2(a2 + b2)/9

Using (1)                                   ab = 2*602/9 = 800

Therefore area of triangle PQR = ab/2 = 400



  Posted by Harry on 2015-12-27 16:45:56
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