Grades 9-10 Video Solutions 2010-Levels 9&10 Video Solutions 2010 problem21

Levels 9&10 Video Solutions 2010 problem21

Video Transcription

Question number 21. In the isosceles trapezoid ABCD, the point X is the midpoint of the segment AB, the segment BX here has length 1, and CXD is a right angle.  What is the perimeter of this trapezoid? So we can write down the expression for the perimeter and then try to fill in the missing information.  So the perimeter, the distance around this trapezoid would be the length of the segment AB plus the length of the segment BC together with AD and finally also we need CD.  Now isosceles implies that the length of AB is equal to the length of CD, so the perimeter calculation is now twice AB plus the length of BC together with the length of AD.  And so we can then focus on the information given in the problem. We know that BX here has length equal to 1. We know that X is the midpoint of AB, so AX also has length equal to 1.  And so this tells us that the length of AB is equal to 2, and so now the perimeter is 2 times 2, so 4, plus the length of BC together with the length of AD.  And because we're looking at a trapezoid, we have the following relationship between the parallel sides BC and AD. The average of those lengths, so we add them and divide by 2, that's equal to the length of the median.  And the median here will be the following line, let me draw that in blue, from X to the point on the opposite side which I will label as Q. So the median here is the length of XQ.  Now, why is XQ the median? That is because X is the midpoint of the segment AB, and drawing a line parallel here to the line segments BC and AD guarantees that CQ will also have length 1 and QD will have length 1, so XQ is exactly halfway from BC to AD and that's why it's the median.  So, solving this equation here we see that 2XQ is equal to BC plus AD and we can use that in our perimeter equation to have 4 plus 2 times the length of the median XQ.  And now we have to know something about the length of the median, and it turns out that because here we have a 90 degree angle, CXQ is a right triangle, and we see from the diagram that XQ bisects that angle, we have two 45 degree angles here in orange.  That tells us that the triangle CXQ is isosceles, so the angle in orange here is the same as the angle here CQX, and that means that the length of the median here is also equal to 1.  So, with that we have 4 plus 2 times the length of XQ which is 1, and so the perimeter comes out to be equal to 6, and that is answer B.

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