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Parallelogram Properties - Problem 3
The diagonals of a parallelogram bisect one another. Recall that this means that they are split into two separate pieces. So, if the bisected segments of the diagonals are given in terms of a variable, set them equal to one another to solve for the variable. Then, add these expressions together. Plug in these expressions to find the length of the diagonal.
We can use what we know about diagonals to solve for missing variables, but what do we know about the diagonals in a parallelogram. Well let’s look at our diagram here.
It look like we have two diagonals that are definitely not congruent to each other, but they are bisected by the other diagonal. So what that means is if I look at this diagonal right here, it is divided into two congruent pieces by the other diagonal, so that’s going to be the key to solving for x and y in this problem where we know that we have two congruent pieces on that diagonal since it’s bisected by the other diagonal and we have two more congruent pieces.
So we can write two equations we can set these two pieces equal to each other. So if I wrote an equation for this diagonal, it would say 2y equals 3x. We have 2 variables, 1 equation that’s not enough information, but we can solve for x in our other diagonal. So I’m going to write that 2x minus 2 is equal to x. Since they are congruent, they have to be equal to each other. So to solve this equation, I’m going to subtract x on both sides 2x minus x is 1x, we get x minus 2 is equal to 0. So if I add 2 to both sides, I find out that x must be 2. So I’m going to come up here and I’m going to write x equals 2.
Now to find out what y is, I’m going to substitute in for x. So I’m going to write that 2y is equal to 3 times 2, 3 times 2 is 6, so we have 2y equals 6 and when I divide by 2, I find out that y must be 3.
The key to solving this problem is to realize that in a parallelogram, the diagonals are bisected by each other.