Easy math puzzles

Pen or pencil, grid paper (cm*** markings/spacing), ruler (cm*** markings/spacing) and a math compass

As long as the grid paper markings and ruler markings are the same, then you can use any unit of measurement, eg inches, mm, etc...

Surely the Wiz doesn't expect us to have all this equipment lying around, to solve an "easier than normal" problem, so there must be a better way (I just can't think of it).

was I at least correct that this is the equipment needed for one way of getting the answer/ approximate answer (or am I wrong?)

Graph paper would certainly be useful to get an estimate, but for full credit, I want to see an algebraic solution.

See image : https://imge.to/i/yV3aUG

From blue triangle, cos(θ) = x/4,

and

From red triangle, cos(θ) = a/3

Therefore, a/3= x/4, a=3x/4, and b = x/4


b^2+x^2 = 4^2

(x/4)^2 + x^2 = 16

x= 16/(17)^0.5 = 3.8806

Note: I know that this formula works for at least one type of "area of a triangle inside a square " problem, but I don't know if it works for the type you posted.

Length of side "X" = Square root of the "area of the square" =
Square root of ("Area of the Triangle" x 2)

Using the above, for a triangle where: 3 = base and 4 = height , then X would be = 3.46... (or square root of 12 to be exact).

X = √ 12 (where X must be a positive number )

I hope this is correct, as I will not be attempting any more answers to this problem if it is wrong (in other words, "too hard for me")

The angle of the 3-4-5 triangle is a right angle, so the angle of the blue triangle at 34 and the red triangle add up to 90. So the blue triangle and red triangle are similar.
Thus if the base of blue is x, the base of red is 3/4 x. So the side of blue is 1/4x.
(Academically you can now work your way round to show the all parts clockwise are x 1/4x 3/4x 3/16x 13/16x x.)

Looking at blue triangle x^2 + (1/4 x)^2 = 16 = 17/16 x^2. So x^2 = 256/17.

If you do the same maths with the others you get the same result (e.g. 25 = (169+256)/256 x^2 = 425/256 x^2).

X = SQRT (256/17).