Answer: Let the two legs of the right triangle be a and b. Then, using the Pythagorean theorem, we have:
a^2 + b^2 = 10^2 = 100
The perimeter of the triangle is:
a + b + 10 = 24
Solving for a and b in terms of one variable, we get:
a = 24 - b - 10 = 14 - b
b = 24 - a - 10 = 14 - a
Substituting these expressions into the Pythagorean equation, we get:
(14 - b)^2 + b^2 = 100
Expanding and simplifying, we get a quadratic equation:
2b^2 - 28b + 84 = 0
Dividing by 2, we get:
b^2 - 14b + 42 = 0
This quadratic can be factored as:
(b - 7)(b - 6) = 0
Therefore, b = 7 or b = 6.
If b = 7, then a = 14 - 7 = 7. If b = 6, then a = 14 - 6 = 8.
So the other two sides of the right triangle are 7 cm and 24 - 7 - 10 = 7 cm, or 6 cm and 24 - 6 - 10 = 8 cm.
Using the formula for the area of a right triangle, we have:
Area = (1/2) * base * height = (1/2) * 7 * 7 = 24.5 cm^2 (for the triangle with legs 7 cm and 7 cm) or (1/2) * 6 * 8 = 24 cm^2 (for the triangle with legs 6 cm and 8 cm)
To verify the result using Heron's formula, we need to calculate the semi-perimeter of the triangle. It is:
s = (7 + 7 + 10)/2 = 12
Using Heron's formula, the area is:
sqrt(s(s-7)(s-7)(s-10)) = sqrt(12552) = sqrt(600) = 24.49 cm^2 (for the triangle with legs 7 cm and 7 cm) or sqrt(12642) = sqrt(576) = 24 cm^2 (for the triangle with legs 6 cm and 8 cm)
As we can see, the results obtained using the two methods are very close to each other, which confirms the correctness of our solution.
Explanation: