Grades 9-10 Video Solutions 2013-Level 9&10 Video Solutions 2013 problem13

Video Transcription

Question number 13, the points P and Q are opposite vertices of a regular hexagon and the points R and S are the midpoints of opposite edges as shown.  The area of the hexagon is 60 centimeters squared.  What is the product of the lengths of the segments P, Q and R, S? Let's begin by expressing the area 60 centimeters squared as the area of six isosceles triangles,  which I will shade in red.  Here is one of them.  And let's do one more.  We know that the area of a triangle is given by 1 half times its base times its height.  So let's look for the base and for the height.  In yellow, I'll shade in the base of one of those isosceles triangles. And we see that it is exactly equal to 1 half of the length  of the line segment from P to Q. We have here a parallelogram so those edges in yellow have equal length.  And in green, I will shade in the height of one of those triangles. And we see that that is exactly half the line segment R, S.  And now we can fill that in our equation. So we have 6 times 1 half.  The base is half the length of the segment P, Q. The height is half the segment R, S. And so we have 6 eighths times the product  P, Q times R, S, which is the quantity we're looking for.  So 60 centimeters squared is equal to 3 fourths  of the product of the lengths P, Q and R, S, which gives us  that 80 centimeters squared is equal to the quantity we are looking for.

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