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Parallelogram Properties - Problem 2
Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
Recall that a parallelogram has two pairs of consecutive sides. From this, it is possible to find the perimeter of a parallelogram some of the sides are given in terms of a variable. First solve for the variable by setting a given side length equal to the expression for the opposite side. Then, substitute this value into the expression for the other side to find its length. Remember that opposite sides are congruent. Also remember that the perimeter of a polygon is the sum of its side lengths. So, add up the value of each side to find the perimeter of the polygon.
One of the key things about geometry is looking at a picture and not assuming anything except for what you’re given. So while this might not look like a perfect parallelogram because I’m not a great artist, we can assume that we have two pairs of parallel sides and we’re being asked to find capital P, which is your perimeter.
So to do that, we’re going to need to know some information about parallelograms. The first thing that’s going to help us is that we know that opposite sides are always congruent in a parallelogram. So what I can do since I don’t know what x is, is I can say that 20 is equal x plus 3 and I know that 2x is going to be this side over here.
Now why am I thinking about all 4 sides? Because we’re being asked to find the perimeter which means I’m going to need to sum the 4 sides of my parallelogram. So if I know what x is, then I can substitute in here and find out what my total perimeter is.
So let’s write an equation that says 20 is equal x plus 3, since opposite sides in a parallelogram are congruent. Well this is a pretty easy equation to solve. We say that x must be 17. Since x is 17, that makes this side 2 times 17, so I guess what I could do is I could write since I have a little more room over here, this is 2 times 17 which is equal to 34.
So to kind of clean this up I’m going to redraw my parallelogram a little bit smaller down here and we see that this is 20, we know that’s 20. Opposite sides are congruent which means that this side is 20. We know that this side is 34 and since opposite sides in a parallelogram, so I should probably mark this as being parallel, 2 pairs of parallel side. We know that this side has to be 34.
So the sum of these, we see 20 plus 20 is 40, plus 34 and 34 is 68, so our perimeter is equal to 60 plus 48 which is 108, and it doesn’t look like we have units in here, so I’m just going to write P is equal to 108 units.
The key thing here was remembering that opposite sides in a parallelogram are congruent.
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