Problem: Find the area of a right triangle whose perimeter is 24 cm and hypotenuse is 10 cm.

Solution:

To solve this problem, we will use the Pythagorean theorem and the formula for the perimeter of a triangle.

Step 1: Determine the lengths of the other two sides of the triangle.

Let's assume that the two legs of the triangle have lengths a and b. Since the hypotenuse is 10 cm, we can use the Pythagorean theorem to find the values of a and b:

a^2 + b^2 = 10^2
a^2 + b^2 = 100

Step 2: Use the formula for the perimeter of a triangle to find the value of a + b.

The formula for the perimeter of a triangle is:

perimeter = a + b + c

where c is the length of the hypotenuse. In this case, we know that the perimeter is 24 cm, and we know that c = 10 cm. So we can substitute these values into the formula and solve for a + b:

24 = a + b + 10
a + b = 14

Step 3: Solve the system of equations to find the values of a and b.

We have two equations:

a^2 + b^2 = 100
a + b = 14

We can use substitution to solve for one of the variables. For example, we can solve for b in terms of a:

b = 14 - a

Then we can substitute this expression for b into the first equation:

a^2 + (14 - a)^2 = 100

Expanding the square and simplifying, we get:

2a^2 - 28a + 96 = 0

Dividing by 2, we get:

a^2 - 14a + 48 = 0

This quadratic equation factors as:

(a - 6)(a - 8) = 0

So the solutions are a = 6 and a = 8. Since a and b are interchangeable, we can also have b = 6 and b = 8.

Step 4: Calculate the area of the triangle.

Now that we know the lengths of the legs of the triangle, we can use the formula for the area of a right triangle:

area = (1/2)ab

Using the values of a and b, we get:

area = (1/2)(6)(8) = 24 cm^2

Therefore, the area of the right triangle is 24 square centimeters.

Perimeter of the triangle =24 cm,
AB+BC+AC=24,
a+b+10=24,
a+b=14  (1),
From Pythagoras theorem,
AB2+BC2=AC2,
a2+b2=102,
a2+b2=100 (2),
Now,
(a+b)2=a2+b2+2ab,
From (1) and (2),
(14)2=100+2ab,
2ab=196−100,
2ab=96,
ab=48,
Area of the triangle =21​×base×height,
=21​ab
=248​
=24  cm2