Take Perpendicular Distances, Get One Side (Posted on 2007-07-08)

(A) In a right angled triangle PQR, an altitude RS is drawn from the vertex R of the right angle. The respective perpendicular distances of the point S from the sides PR and QR are 6 and 3.

Determine the length of PR.

(B) What would have been the length of PR if the respective perpendicular distances of the point S from the sides PR and QR were 8 and 4?

Let the respective distance of S from the sides PR and QR be s and t. Draw ST perpendicular to QR and QU perpendicular to RP.
Then, ST = a (say) and SU = b (say).

We also observe that USTR is a rectangle. Let PR = c
Then, from the similarity of triangles, we obtain:

RU/ SU = SU/PU
Or, SU^2 = RU*PU
Or, b^2 = a*(c-a)
Or, c = (a^2 + b^2)/a

For PART (A), substituting (a, b) = (3, 6), we obtain:

PR = (3^2+6^2)/3 = 15

For PART (B), substituting (a, b) = (4, 8), we obtain:

PR = (4^2+8^2)/4 = 20

....................QED......................

*** Also refer to an alternative methodology provided by Praneeth Yalavarthi in this location.

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