If 𝐹𝐽 is equal to six 𝑥 plus eight 𝑦, 𝐹𝑀 is equal to three 𝑥 plus five 𝑦, 𝐺𝐻 equals 42, and 𝐺𝑀 equals 24, what values of 𝑥 and 𝑦 make parallelogram 𝐹𝐺𝐻𝐽 a rectangle?
So we’ve been given a lot of information in the question about different lengths within the diagram. So let’s begin by transferring this information onto the diagram itself. We’re told that 𝐹𝐽 is equal to six 𝑥 plus eight 𝑦. And the opposite side, 𝐺𝐻, is equal to 42. We’re also given some information about parts of the diagonals of this parallelogram, which is that 𝐹𝑀 is equal to three 𝑥 plus five 𝑦 and 𝐺𝑀 is equal to 24.
Now we’re asked what values of 𝑥 and 𝑦 make this parallelogram into a rectangle. So we need to consider what properties a rectangle has that aren’t true of parallelograms in general. Now there are a couple of properties. But seeing as we’ve been given the lengths of part of the diagonals of this parallelogram, the ones that are relevant here are these: firstly, that the diagonals of a rectangle are equal in length; secondly, they bisect each other, which means they cut each other in half.
This means then that all four of the line segments going from a vertex of the rectangle to the centre point 𝑀 are equal in length. Specifically taking the two for which we’ve been given expressions or values, we can write down the equation three 𝑥 plus five 𝑦 is equal to 24. And so we have an equation connecting the values of 𝑥 and 𝑦. However, this is only one equation with two unknowns. And so this isn’t sufficient to be able to solve it.
Let’s think about what other information we’ve been given. Remember in any parallelogram, the lengths of opposite sides are equal. Therefore, the sides 𝐹𝐽 and 𝐺𝐻 are equal in length. So we can write down another equation: six 𝑥 plus eight 𝑦 is equal to 42. Now we have a pair of simultaneous equations in 𝑥 and 𝑦, which we can solve in order to determine the values that make the first equation true and therefore make the parallelogram a rectangle.
In this second equation, all of the coefficients are even numbers, which means the whole equation can be divided by two, giving the simpler equation three 𝑥 plus four 𝑦 is equal to 21. We’ll now look at solving this simpler equation, which I’ve labelled equation two, along with the first equation, equation one. We first noticed that both equations have the same coefficient of 𝑥, it’s three, which means that if I subtract equation two from equation one, this will eliminate the 𝑥 terms.
So if I go ahead and do this, then as intended the 𝑥 terms will cancel out. Three 𝑥 minus three 𝑥 give no 𝑥s. Five 𝑦 minus four 𝑦 gives one 𝑦, which we just write as 𝑦. And 24 minus 21 gives three. So we now have the equation 𝑦 is equal to three. And in fact we found the value of 𝑦. In order to find the value of 𝑥, we can substitute this value of 𝑦 into any of the equations. I’m going to choose to substitute into the equation labelled one.
Making this substitution gives three 𝑥 plus five multiplied by three is equal to 24. Five times three is 15. So we have three 𝑥 plus 15 is equal to 24. Next, I subtract 15 from each side of the equation, giving three 𝑥 is equal to nine. The final step is to divide both sides of the equation by three, giving 𝑥 is equal to three. So we found the values of 𝑥 and 𝑦 that make the parallelogram 𝐹𝐺𝐻𝐽 a rectangle. They’re both equal to three.
Remember, the key facts we used in this question, which are properties of rectangles but not all parallelograms, are these: the diagonals of a rectangle equal in length and they bisect each other.