MAML, Geometric Similarities: A Right Trapezoid
Channel: Noble Mushtak
Date: 2015-07-29

This is a MAML (Maine Association of Math Leagues) problem from October 2011 (Meet 1 of 2011-12), Round 5, Problem 2: http://maml.net/problems.htm

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Condensed Writing Solution:
Draw a line parallel to QR through S and call the point of intersection of PS and this line N.
Quad. NSTQ is a rectangle, so SN is congruent to QT and QN is congruent to ST because opposite sides of rectangles are congruent. Thus, SN=24 and QN=8.
PQ=PN+QN by the Addition Postulate, so 15=PN+8, meaning PN=7.
PN^2+SN^2=PS^2 by the Pythagorean Theorem, so 7^2+24^2=625=PS^2, meaning PS=25.

Angle SNP is congruent to angle STR because they are both formed by perpendicular lines, meaning they are both right and all right angles are congruent.
Because of this and the given that angle SPN is congruent to angle SRT, triangles SNP and STR are similar by the Angle-Angle Theorem.

Thus, by the Corresponding-Sides-of-Similar-Triangles-are-Proportional Theorem, SR/PS=TR/PN=ST/SN=8/24=1/3.
SR/PS=1/3 implies SR/25=1/3 implies SR=25/3.
Similarly, TR/PN=1/3 implies TR/7=1/3 implies TR=7/3.

Thus, the perimeter of Quad. PQRS is PS+SR+TR+QT+PS=25+25/3+7/3+24+15=74 2/3 ANSWER