Archer is sewing a triangular flag. He starts with
a piece of fabric in the shape of a right triangle, with one leg measuring 12 inches and the other leg measuring 9 inches. He plans to trim off the corners of the triangle so that the resulting flag is equilateral. To do this, he will need to remove a triangular piece from each corner of the original triangle.
To figure out how much fabric he needs to remove, Archer first needs to find the hypotenuse of the original triangle. He can use the Pythagorean theorem, which states that in a right triangle with legs a and b and hypotenuse c, c^2 = a^2 + b^2. Plugging in the values for the legs of the original triangle, Archer gets:
c^2 = 12^2 + 9^2
c^2 = 144 + 81
c^2 = 225
c = 15
So the hypotenuse of the original triangle measures 15 inches. To make the flag equilateral, Archer needs to remove a piece of fabric from each corner of the original triangle that is shaped like a smaller equilateral triangle. He can find the size of these smaller triangles by dividing the length of the hypotenuse by 2, since the resulting equilateral triangle will have sides that are half as long as the hypotenuse of the original triangle. So Archer needs to remove equilateral triangles with sides measuring:
15 / 2 = 7.5 inches
To calculate the area of each of these smaller triangles, Archer can use the formula for the area of an equilateral triangle, which is:
area = (side^2 * √3) / 4
Plugging in the value for the side of each small triangle, Archer gets:
area = (7.5^2 * √3) / 4
area = (56.25 * 1.732) / 4
area = 30.836
So the area of each of the small triangles that Archer needs to remove from the corners of the original triangle is about 30.836 square inches. To find out how much fabric he needs to remove in total, he can multiply this value by 3 (since there are 3 corners to trim). So Archer needs to remove a total of:
30.836 * 3 = 92.508
or about 92.5 square inches of fabric to make the equilateral flag.