text: in triangle pqr, ps is the altitude from p to qr, and qs does not equal rs. prove that pq does not equal pr.?   ps is the altitude means that ps is perpendicular to qr. so <psq is a right angle. use the pythagorean theorem.  (qs)^2 + (ps)^2 = (pq)^2 and (rs)^2 + (ps)^2 = (pr)^2 subtract one from the other to get:  equation 1: (qs)^2 - (rs)^2 = (pq)^2 - (pr)^2 so if qs ≠ rs then (qs)^2  ≠ (rs)^2 subtract (rs)^2 from both sides to get: (qs)^2 - (rs)^2 ≠ 0 substitute this into equation 1 to get: 0 ≠ (pq)^2 - (pr)^2 add (pr)^2 to both sides: (pq)^2 ≠ (pr)^2 and since all values are positive, we can take the principle square root of both sides to get: pq ≠ pr