where , , and cm. Find , the length of , and the ratio between the area of and that of .
In and , corresponds to , corresponds to , and corresponds to . Also, side corresponds to side , side corresponds to side , and side corresponds to side . Corresponding angles of similar triangles are congruent, so . Therefore, since measures , must also measure . Use the fact that the lengths of corresponding sides of similar triangles are proportional to determine the length of .
Now use the fact that the ratio of the areas of the surfaces of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides of the triangles to find the ratio between the area of and that of .
Thus, , the length of is , and the ratio between the area of and that of is .