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|Triangle Area Ascertainment (Posted on 2015-12-26)
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.
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Submitted by K Sengupta
Rating: 5.0000 (1 votes)
| Comment 1 of 2
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
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
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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