Final answer:
To find the largest possible area of a right triangle, we can use the formula for the area of a triangle. Given that the sum of the lengths of the two shorter sides is 19 cm, we can find the values of the sides using the Pythagorean theorem. The largest possible area of the triangle is 45.25 square cm.
Step-by-step explanation:
To find the largest possible area of a right triangle when the sum of the lengths of the two shorter sides is 19 cm, we can use the formula for the area of a triangle, which is 1/2 times base times height. Given that the two shorter sides have a sum of 19 cm, we can let one side be x cm and the other side be (19 - x) cm. Using the Pythagorean theorem, we can find the relationship between the sides of a right triangle: a^2 + b^2 = c^2, where a and b are the two shorter sides and c is the hypotenuse. Substituting the values, we get x^2 + (19 - x)^2 = c^2. We can simplify this expression to get 2x^2 - 38x + 361 = c^2. To find the largest possible area, we need to find the value of x that maximizes c^2. Since c is the hypotenuse, it must be the longest side, and therefore, we need to maximize the square of c. This occurs when x = 9.5 cm, which means the two shorter sides have lengths of 9.5 cm and 9.5 cm. The hypotenuse (c) can be found by substituting x = 9.5 into the equation 2x^2 - 38x + 361 = c^2, which gives us c = 10.5 cm. Finally, we can calculate the area of the triangle using the formula 1/2 times base times height, where the base is one of the shorter sides (9.5 cm) and the height is the other shorter side (9.5 cm), giving us an area of 45.25 square cm.
.