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Isosceles Leg Length (Posted on 2016-06-09) Difficulty: 3 of 5
An isosceles triangle PQR is such that ∠PRQ = 90o. Point U is inside triangle PQR, such that UP =11, UQ=7 and UR=6.

Each of the legs PR and QR has length √(X/Y) where each of X and Y is a positive integer.

Find X and Y

No Solution Yet Submitted by K Sengupta    
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Some Thoughts A problem | Comment 1 of 6

Construct line AB, and line CD perpendicular to it. Their intersection is R.

Construct circle O1, radius, 6 on R. U is a point on O1 in the first (positive) quadrant of AB/CD.

Construct circles O2, radius 11, and O3, radius 7, on U. These cross AB and CD at P and Q respectively.

Since each of the legs PR and QR has length √(X/Y), they are the same length. This occurs when PQ is near 17. That would give the answer X/Y= 289/2.

However, it seems that PQ may be slightly less than 17, and closer to 16.994, giving X/Y a value very close to 7599289/632403.

Alternatively, each of PR and QR could have a different length expressible as the square root of a rational fraction, but it seems that in that case multiple solutions are possible.



Edited on June 10, 2016, 2:06 am
  Posted by broll on 2016-06-10 01:40:25

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