Question number 17, in the quadrilateral PQRS shown below, which line segment is the longest? So the line segments here are the edges of the two triangles that make up this shape. Let's decide first of all that we only need to focus in on the longest sides from each triangle and these are the sides opposite the largest angles. So let's fill in the angle measures that are missing. We have here a 61 degree angle and a 59 degree angle and so the sides that we will focus on are the longest sides and those are the segments PS and RS. Now let's study them closely by writing down an expression using similar triangles. So by similar triangles we have the ratio PS to RS being the same as SQ compared to RQ. And I will argue that that number is bigger than 1. We do that by again deciding that SR is the length of the edge opposite the largest angle so it must be longer than SQ which is opposite the 60 degree angle and that has to be longer than RQ which is opposite the smallest angle. So we have that SQ over RQ is bigger than 1 and that tells us that PS over RS is bigger than 1 or equivalently that after multiplying through PS is longer than RS and those 2 edges being the longest, PS must be the longest of them all.

quadrilateral

longest segment

triangles

similar triangles

PS