### Brain Teasers

# Square Flag

A square flag has a blue field with two white diagonal bars that form a cross. Each bar extends across the entire diagonal, from corner to corner. The cross occupies half of the flag's area. The flag measures 60 inches on each side. To the nearest 0.01, what is the width of the bars?

### Answer

12.43 inchesa) The square flag measures 60 inches on a side, so its area is 60*60 = 3600.

b) The white cross occupies half of the area, or 3600/2 = 1800.

c) Since the flag is square, the cross divides the flag's blue field into four equal-sized isosceles right triangles. Each of those triangles has an area of 1800/4 = 450.

d) The area (A) of an isosceles right triangle is calculated by the formula 0.5(X^2) = A, where X is the length of one of the two perpendicular legs. 0.5(X^2) = 450, or X^2 = 900, or X = 30. These perpendicular legs form the border between the field and the cross.

e) The hypotenuse (H) of an isosceles right triangle is calculated by the formula H = (2(X^2))^(1/2), where X is the length of one of the two perpendicular legs. H = (2(30^2))^(1/2) = (2(900)^(1/2) = 1800^(1/2) = 42.43.

f) The hypotenuse of each triangle lies along the perimeter in the center of each side of the flag. The remaining length of each side is split equally between the two bars, leaving (60-42.43)/2 , or 8.79, for each bar.

g) Where the bars extend into the corners of the flag, isosceles right triangles with legs of 8.79 are formed. Again, use the formula H = (2(X^2))^(1/2). H = (2(8.79^2))^(1/2) = (2(77.26))^(1/2) = 154.52^(1/2) = 12.43 = H. The hypotenuse of 12.43 is the width of the bars.

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