You guys. Crazy story. I was in a coffee shop messing around with Geometer’s Sketchpad, just slapping together some perpendicular line segments

because that’s how I roll, and this guy who could see my screen thought I was an artist or something. It was kinda weird, but he wanted to buy my “art.” Naturally, I went along with it. So now I have to frame this stupid thing and deliver it to him. Thing is, he wasreallyparticular about how it should be framed. He wants both pointsAandJto be on one of the rectangular frame’s diagonals, and a distance of exactly 4 cm along those diagonals fromAandJto the inner corners of the frame. He wants the frame to be 3 cm thick, too. Anyway, he’s busy clearing space for this thing in his house and he’s emailing me asking for dimensions and basically driving me crazy. Can you help me out? What will be the outer dimensions of the framed piece of art? I’ve included measurements of all the lengths of the segments, but don’t count those as part of my image—I’m going to delete them before I print this out.

Post your answers—rounded to 2 decimal places—in the comments. The prize for the first correct answer: I will name my next pet goldfish after you. Scout’s honor.

**UPDATE:** Here’s where I sheepishly admit that this is not one of my best challenges. Sorry. But Rob finally did get it, and even though I didn’t plan to when I posted this, I’m gonna send him a Math Guide for his troubles.

Solution below the cut.

The first thing you’re gonna want to do is construct a right triangle out of this bad boy, which means you’re going to need to find the total vertical length and the total horizontal length. What a pain!

Vertical length (*AK*) = *AB* + *CD* – *EF* – *GH* + *IJ* = 12.87

Horizontal length (*JK*) = BC + DE + FG + HI = 12.88

So that’s super annoying and why this is not one of my best challenges—Geometer’s Sketchpad is an awesome program that I love and it makes making diagrams really easy and fun, but this was drawn up to be a 45º-45º-90º triangle and because of rounding now it’s not quite so pretty. We’ll be OK, but still. Annoying.

Anyway, now we can find *AJ*, the hypotenuse.

12.87^{2} + 12.88^{2} = *AJ*^{2}

331.5313 = *AJ*^{2}

18.2080… = *AJ*

OK, good enough. We’re told in the problem that the inner corners of the frame are to be 4 cm from *A* and *J*. How the heck do we do that?

Easy, friend. *Similar triangles*. Picture it with me. You’re going to be creating another, larger, right triangle when you extend that hypotenuse by a total of 8cm. So with a few quick ratios, we can determine the inner dimensions of the frame!

The hypotenuse of our large triangle is going to be 18.2080… + 8 = 26.2080…

Do some ratios to figure out the height and width:

But we’re almost done. Now all we need to do is account for the width of the frame, which is supposed to be 3 cm. Because the frame’s thickness will add to top *and* bottom, left *and* right, we add 6 cm to each side to arrive (finally) at our outer dimensions: 24.52 cm by 24.54 cm.

## Comments (6)

this is what i cud think of . the language of the question adds to the mystery. In all probabilities this is wrong,but maths is all about screwing it up and messing things up royally !

24.54 cm (wide) x 24.53 cm (long)

Sorry meant 24.54 cm (long) x 24.53 cm (wide)

I don’t like my answer…is this supposed to be a square frame? I keep getting internal dimensions to be 12.88cm and 12.87 cm…my answer is wrong I believe if these are the right dimensions but if it’s a square frame the answer will be different and easy to get, just need to know if 12.88 or 12.87 is correct….

Never mind I dumb…you said rectangular…

It’s funny. My intention in designing this problem was to make the frame a square. I tweaked the lengths until the angle BAJ was exactly 45º. But then the rounding ended up being off, as you point out, which makes it more difficult. I’ll post a solution soon.