Isosceles Triangles Noah

There are 12 possible triangles you can make with these criteria. To get an area of 9 cm^2, the base*the height*1/2 has to equal 9. Multiplying the equation by 2 on both sides gives bh=18. In an isosceles triangle, where a and b are the bottom vertices and c is the top vertex, the distance between a and c = the distance between b and c. Therefore, the x coordinate of a + the x coordinate of b divided by 2= the x coordinate of c. a is the point 20,20. Hence, the x coordinate of b has to be even because if not, 20+k (where k is odd) divided by 2 would not be whole and the x coordinate of c would also not be whole. Therefore, the x coordinate of b has to be 20 + any even factor of 18. That means the x coordinate of b could be 22, 26, or 38. The coordinates of b could be 22,20 or 26,20 or 38,20. To figure out the x coordinate of c, we do 20+22/2 or 20+26/2 or 20+38/2= 21, 23, or 29 to figure out the x coordinate. To figure out the y coordinate of c, we do 20+ 18 divided by the "even factor of 18" part from the x coordinate of b. This yields us 21,20+(18/2) , 29,20+(18/6) , 29,20+(18/18)= 21,29, 23,23 and 29,21. So the triangles that we get as a result are a=20,20, b=22,20 and c=21, 29 and a=20,20, b=26,20, c=23,23 and a=20,20, b=38,20 and c= 29,21. Accounting for reflections, there are 9 more triangles like these. 3+9=12 so there are 12 triangles.