Answer: Let the legs of the right triangle be a and b. We know that the radius of the inscribed circle is 4 cm, which means that the inradius is also 4 cm. The inradius is given by:
r = (a + b - c)/2
where c is the hypotenuse, so we can substitute the given values to get:
4 = (a + b - 35)/2
Multiplying both sides by 2, we get:
8 = a + b - 35
Adding 35 to both sides, we get:
a + b = 43
We also know that the area of the triangle is given by:
A = (ab)/2
where a and b are the legs of the triangle. We can use the Pythagorean theorem to relate the legs to the hypotenuse:
a^2 + b^2 = c^2
Substituting the given value for the hypotenuse, we get:
a^2 + b^2 = 35^2
Simplifying, we get:
a^2 + b^2 = 1225
Now we can use the equation for the sum of the legs to solve for one of the legs in terms of the other:
a + b = 43
b = 43 - a
Substituting this expression for b into the equation for the area, we get:
A = (a(43 - a))/2
Simplifying, we get:
A = (43a - a^2)/2
To find the maximum value of the area, we can take the derivative of A with respect to a and set it equal to 0:
dA/da = 43/2 - a = 0
Solving for a, we get:
a = 43/2
Substituting this value into the equation for the area, we get:
A = (43/2)(43/2 - 43/2)/2 = 0
This means that the area of the triangle is 0 when one of the legs has a length of 0, which is not possible. Therefore, the maximum area occurs when the legs have equal lengths, which means that the right triangle is an isosceles right triangle. In this case, we have:
a = b = (35/sqrt(2)) cm
Substituting these values into the equation for the area, we get:
A = (ab)/2 = ((35/sqrt(2))^2)/2 = 612.5 cm^2
Therefore, the area of the right triangle is 612.5 square centimeters.
Explanation: