Sigh... I'm doing independant study to get some much needed math credits and I'm starting to get a feeling that the work material was designed for the sole purpose of testing my sanity. The key questions (they're like mini-tests) at the end of each lesson aren't so much fair as they are mind blowing curveballs from math- . It's like the people who wrote the lessons gave up at the end and figured that instead of thinking up challenging relevant questions, they'd just toss in a bunch of random nonsense that's impossible to answer given what's actually taught in the lesson.
Oh well, enough complaining. I'm determined, and it's true that you can't keep a good man down. All that's needed is a little help from you, the hero. The one who's going to explain to me how I can prove quadrilateral W(-0.5, 1.5) X(4, 1) Y(3.5, 5.5) Z(-1, 6) is infact, a rhombus.
If you think that this is a difficult problem, you ain't seen nothing yet!
It's always a good idea to make a sketch . . .Prove that quadrilateral: W(-˝, 1˝), X(4, 1), Y(3˝, 5˝), Z(-1, 6) is a rhombus.
Code:Z | * | Y (-1,6) | * | (3˝,5˝) | | | W | * | X (-˝,1˝)| * | (4,1) - - - - + - - - - - - - - - - |
Do you know what makes a quadrilateral a rhombus?
. . That's right . . . it has four equal sides.
Does WXYZ has four equal sides?
. . How can we determine if: . ?
That's right . . . the Distance Formula.
Go for it!
NB: After you’ve made your sketch, as Soroban did, you do not have to use the distance formula for all four sides! Just use the distance formula on any two adjacent sides, say WX and WZ. Then all you need to do next is show that the vectors and are equal, i.e. the sides WX and ZY are equal and parallel. That is all.
Everyone here seems so one-tracked about this problem that they seem to have forgotten that the rhombus has other properties too. Recall that a rhombus is a parallelogram. So another way of proving that a quadrilateral is a rhombus is to show that it is a parallelogram with equal sides. Showing that it has four equal sides is only way – but not necessarily the easiest way. Don’t be too narrow-minded when you are solving math problems.
In fact, you can also show that it’s a rhombus without using the distance formula and messing around with squares and square roots at all. Once you’ve shown that and , the fact that the other two sides are also equal and parallel will follow; hence WXYZ is a parallelgram. To show that it’s a rhombus, take the dot product of and and show that it is 0, i.e. the diagonals are perpendicular. The rhombus is the only parallelogram whose diagonals intersect perpendicularly.