Geometry problem (NOT homework)

I assure you that this is not a homework problem, I’m a nearly 40-year-old computer programmer just trying to measure plywood to cover a window to protect it from hurricanes.

Please see this line-art drawing of the window:

I have all the outer measurements (in the larger font), and 25" is the measurement from the bottom leftmost corner to 25 inches in, to the first red line.

I do not remember how to figure out how long the first red line is.

I’d simply make the whole thing and then split it into thirds afterwards, but the plywood sheets aren’t big enough to do this, I have to make the “thirds” from scratch.

I tried to find the solution to this on Geometry web sites, but no luck.

First red line shall be designated as r.

The section of r above the horizontal line shall be designated as x.

Then you have:

r = 37.5 + x

The right side of the triangle is 62.5 - 37.5 = 15.

This gives us the equation 15/x = 74/25, or x = 15:(74/25) = 5.07

r = 37.5 + 5.07 = 40.07

Okay, first off. All three acute triangles sharing that narrow angle are similar. (with their right edges being segments of the two red lines, and the rightmost vertical black line in the figure.) Therefore, the ratio of their corresponding edges will always be a constant.

For the largest triangle, the ratio of height to base is 25:74 (25 being 62.5 - 37.5) Since the smallest triangle’s base is also 25, its base, (which is the portion of the leftmost red line that sticks up above the black horizontal line) will be 25/74*25 or about 8.45

The bottom section of both red lines are 37.5 long, so the first red line will be a total of about 45.95 inches long.

This is assuming that the quadrilateral on the bottom is a perfect rectangle, of course, which it seems to be.
on preview: schnitte, I think you miscalculated 62.5 - 37.5 and lost ten.

Looks like a “similar triangles” problem. Or maybe even just a Pythagorean Theorem problem.

The red line has two pieces, the part that’s a side of a rectangle (which is 37.5 inches long) and the part that’s a side of a triangle (which we will call x inches long). The red line is 37.5 + x inches long. So, we need to know what x should be.

The large triangle (with a hypotenuse of 78 inches, a base of 74 inches, and a height of 62.5 - 37.5 = 25 inches) is “similar” to the smallest triangle, meaning that the ratio of two sides of one of these triangles will be the same as the ratio of the corresponding two sides of the other triangle.

Thus, take the ratio of height to base. For the large triangle, this ratio is 25/74. For the small triangle, this ratio is x/25. So 25/74 = x/25. Multiply both sides by 25, and you get that x = 625/74. Pulling out my calculator, I approximate that as 8.4459… inches.

So that first red line is 37.5 plus about 8.4 inches, for a total of 45.9 inches.

Consider that strength may be a more important factor than efficient use of plywood but I’m pretty sure you can cover this with a single sheet without doing any calculations. The more cuts you have in the wood the weaker it will be. If I was covering a window I’d start with a 4’x8’ sheet laid out horizontally. Cut off the end at 74" and use the cutoff to cover the bare triangle at the top.

Thanks guys, I totally forgot about the similar triangles thing. In fact, I still don’t remember it, even though I did pretty well in geometry all those years ago. Too many years, I guess.

Padeye, that’s a good point about the strength, but I forgot to mention this is 2nd floor window, and I also want to break it up just because it’s easier to carry up smaller pieces than one big piece.

Just to verify I’m understanding this properly:

Suppose the second red in 50 inches from the left edge. Let’s say y is the portion of the second red line above the rectangle.

length of the the red line is 37.5 + y

y/50 = 25/74

hence the second red line is 37.5+16.89 = 54.39

is this correct?

yeah, that all checks out by me.

Thanks again.

I did. Thank you.

A few calculators that really come in handy for this are:

(Needless to say, I agree with the 8.45 calculation too).