Question number 22. Lines parallel to the base of an equilateral triangle divide both its sides into 10 equal pieces. So the little segments here between the shaded and unshaded strips are of the same length and 10 of them exactly make up the length of the side. So one of them would have length s, let's say, and then the whole side is 10s, so here the remaining pieces would be together 9s and the third side also measures 10s. So the question is what percentage of the triangle is shaded? Well, to figure that out we would have to have the following ratio, the shaded area divided by the total area, and at least the total area we can compute, so we use the formula 1 half a b sine of the included angle c and that would give us 1 half times 10s times 10s times the sine of the 60 degree angle, since this is a nice equilateral triangle, and that gives us 1 half quantity 10s squared times root 3 over 2. So simplifying a little bit we have in the denominator here the total area is root 3 over 4 and then 10s quantity squared, and I'll just leave it like that for now. Now for the shaded area what we will do is we will compute the area of the triangle here with side length equal to 9s, subtract from that the area of the triangle with side length equal to 8s, then add back to that the area of the triangle with side length equal to 7s and so on until we subtract the area of the triangle here with side length equal to 2s and add back in the last piece which has side length equal to 1s. So let's set that up. What we would have is for the shaded area computation we have root 3 over 4 and then the side length here is 9s, so that's squared, subtract from that root 3 over 4, 8s quantity squared, add back root 3 over 4, 7s quantity squared, and we continue until we subtract root 3 over 4 quantity 2s and finally add back in root 3 over 4, 1s quantity squared. So this is equal to, we can factor out root 3 over 4, we can factor out s squared, and what remains is 9 squared minus 8 squared plus 7 squared minus 6 squared plus 5 squared minus 4 squared plus 3 squared minus 2 squared plus 1 squared and that we can compute root 3 over 4 s squared remains outside and then we have 81 minus 64 which is 17, we have 49 minus 36 which is 13, we have 25 minus 16 which is 9, next is 9 minus 4 so 5 and then plus 1 squared that's 1 and that gives us in the brackets the number is 45. So up top in the numerator in our ratio we have root 3 over 4 and then s squared here and times 45, so the root 3 over 4 cancels, s squared cancels, and what we end up looking is 45 divided by 10 squared so divided by 100 and that is 45 percent. So 45 percent of the area here is shaded and we choose answer C for question number 22.

equilateral triangle

shaded area

area calculation

geometry problem

percentage