Saturday Puzzler -- Triangle and Square

Re: Saturday Puzzler -- Triangle and Square

Larry Barrett

1 - 1/2 - 1/6

Correct!

Alex Y

I obviously have to find some tougher puzzles!

Just curious if 1 - 1/2 - 1/6 was just your way of giving the answer of if it is how you solved it. If the latter, how did you come up with 1/6?

The solution I saw noted that the triangles created on the bottom and top sides were similar. Taking the horizontal sides of the triangles as the bases, we see a 2:1 ratio, that also applies to the altitude, so the altitude of the larger of the triangles is 2/3 times the base.

Area of triangle = 1/2 * base * altitude

= 1/2 * base * 2/3 * base

= 1/3 * base^2

Re: Correct!

Henry Higginbotham

That's more elegant than how I saw it, which is that one line has a slope of 1 and the other a slope of -2, so their intersection must give the triangle a height of 2/3. But by then Larry had already posted the answer.

Re: Correct!

Larry Barrett

My way probably more complicated than your's or Henry's solutions.

Label the corners of the square, starting in NW corner and moving CW: A, B, C, D Label each side s and label the midpoint of side AB F.

Draw a diagonal from D to B.

Draw another line from C to F and label the intersection of this line with line DB E.

We want to know the area of triangle DEC with respect to the area of the square, which is s^2.

Area of DEC = s^2 -area of DAB - area of CEB.

Area of DAB = (1/2)s^2.

Area of CFB = (1/2)bh = (1/2)((1/2)s)s = (1/4)s^2.

Area of CFB also = area of EFB + area of CEB.

Let x be the altitude of EFB (from E to line FB). Observe that x is also the altitude of CEB since each altitude is perpendicular to the perpendicular sides and each is drawn from a point (E) on the diagonal to each of the perpendicular sides.

So 1(/4)s^2 = (1/2)((1/2)s)x + (1/2)sx = (3/4)sx

So x=(1/3)s

So area of CEB = (1/2)s(1/3)s = (1/6)s^2.

Therefore area of DEC = s^2 - (1/2)s^2 - (1/6)s^2.

Re: Correct!

Alex Y

Henry, I actually think your solution is the most elegant. Many paths to the same result!