Okay … c’mon now here people! Let’s get into the geometry spirit of the thing! I’m always lecturing my geometry students that they must go by the labels given, and not by appearances. (And yet I do want them to recognize appearances too when it’s appropriate to do so.) But that is easy compared to going strictly by what’s labeled, which is hard when your eyes tell you what it looks like. It must be sort of like a novice pilot being trained for a commercial license who is told to trust the instruments instead of his senses. It’s how you do proof.
Toward the end of the year, my geometry exams may include pictures of quadrilaterals (visually indistinguishable from squares - in fact they may be squares) with various information included about them, such as: all you can see about this quadrilateral is that one of its diagonals bisects the other, and it has one pair of parallel sides.
Question: What is the most specific kind of quadrilateral you can prove any quadrilateral must be in order to meet these criteria?
Multiple choice:
quadrilateral, trapezoid, isosceles trapezoid, parallelogram, rhombus, rectangle, square.
(I made this a tougher one than most of them I throw on an exam - so I had to think about this one for a bit. Let your guesses be known if you dare. And be prepared to prove it to skeptics.) I guess the humor of all this can just be that some of us think math stuff is fun.
I knew there must be an angle and this one is a fairly cute one.
From the markings we can only assume two sides are parallel so it is at least a trapezoid. One diagonal is bisected by the other. So could that happen unless it is at least a parallelogram? I can’t see why a trapezoid being isosceles would ensure it. The right triangles formed by the bisectors and the altitude connecting the midpoints of the two identified parallel sides would be similar but unless they are congruent, not only would the bisectors not have to divide each other into two equal lengths, I don’t think they could do so because similar triangles only have corresponding sides that are equal if the triangles are congruent. So it must be at least a parallelogram so rectangles including squares as special parallelograms work too.
Goid one, Merv!
Full disclosure I never taught grometry so don’t assume what I said is correct unless Merv signs off on it.
Right you are! (You, but not necessarily the triangles, that is!) And while it could be a square, it doesn’t have to be. What it has to be is a parallelogram, as a modified picture of it shows. Had I included points to identify in the first picture, you could probably have explained it more easily. My own proof involved establishing that the top and bottom triangles are congruent (using alternate interior angles on the parallel lines, and then AAS) - which then leads to the conclusion that the other pair of triangles must be congruent - SAS and the other pair of lines must also then be parallel (congruent alternate interior angles for those) … hence … parallelogram.
I like math (I have a degree in it ; - ) but it’s been 60 years since I’ve done any proofs in plane geometry! I’m way happier with analytic. Proofs in trig might be an issue too though.
Yeah I approached it round aboutly by focusing on whether an isosceles trapezoid would/could account for a bisected diagonal formed by its intersection with the other diagonal. Just goi g straight for the parallelogram is more elegant.
Yep! The only isosceles trapezoid that could make even just one diagonal bisect the other would have to be a square. Or another variation: had I added a right angle marker where the diagonals intersect at the center - then ‘square’ [Rhombus] would have been the only qualifying answer. Isosceles trapezoids generally are an intesting shape. Oh - and I had forgotten to add ‘kite’ to my original multiple choice list. Another interesting quadrilateral.
But meanwhile … to segway back into humor where this thread belongs …
What do you call it if you’ve fed your parakeet poisoned crackers?
another oldy but goodie (probably already in this thread somewhere…
[and speaking of diagrams not drawn to scale … check out the relative given measurements!]
I didn’t realize the Library of Congress had stopped saving every tweet. I guess it’s important that we maintain a record of how it all went wrong. Still, I’m sure the world wide web abhors a vacuum about as much as nature does – something else will come along and take Twitter’s place if it comes to that.
Who among us would cherish the thought of our every word spoken as a kid or immature junior high student (or older!) all being recorded for posterity? That’s probably what Twitter is for our society.
[From a non-Twitter user … yeah I know, plenty of responsible reporters and such were obliged to hang out there to stay on top of breaking news. Even cess pools must be attended I suppose.]
Yes, what shall the future generations do without countless images of people’s lunch, videos of their cats running from cucumbers, and links to their blogs.
Goid one, Merv!
Still, I’m sure the world wide web abhors a vacuum about as much as nature does – something else will come along and take Twitter’s place if it comes to that.
