The perimeter of a right angled triangle is 24 cm. If its hypotenuse is 10 cm then area of this triangle is

Options

A. 24 `cm^2`
B. 10 `cm^2`
C. 12 `cm^2`
D. 48 `cm^2`

Answer

To solve the problem step by step, we need to find the area of a right-angled triangle given its perimeter and hypotenuse. Here’s how we can do it:

### Step 1: Understand the given information
We are given:
- Perimeter of the triangle (P) = 24 cm
- Hypotenuse (h) = 10 cm

### Step 2: Set up the equation for the perimeter
The perimeter of a triangle is the sum of all its sides. For a right-angled triangle, we can denote the two other sides as \( a \) and \( b \). Therefore, we can write the equation:
\[ a + b + h = P \]
Substituting the known values:
\[ a + b + 10 = 24 \]

### Step 3: Simplify the equation
From the equation above, we can isolate \( a + b \):
\[ a + b = 24 - 10 \]
\[ a + b = 14 \]

### Step 4: Use the Pythagorean theorem
For a right-angled triangle, the Pythagorean theorem states:
\[ h^2 = a^2 + b^2 \]
Substituting the value of the hypotenuse:
\[ 10^2 = a^2 + b^2 \]
\[ 100 = a^2 + b^2 \]

### Step 5: Set up a system of equations
Now we have two equations:
1. \( a + b = 14 \) (Equation 1)
2. \( a^2 + b^2 = 100 \) (Equation 2)

### Step 6: Express one variable in terms of the other
From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 14 - a \]

### Step 7: Substitute into the second equation
Now substitute \( b \) into Equation 2:
\[ a^2 + (14 - a)^2 = 100 \]

### Step 8: Expand and simplify
Expanding the equation:
\[ a^2 + (196 - 28a + a^2) = 100 \]
Combine like terms:
\[ 2a^2 - 28a + 196 = 100 \]
Subtract 100 from both sides:
\[ 2a^2 - 28a + 96 = 0 \]

### Step 9: Simplify the quadratic equation
Dividing the entire equation by 2:
\[ a^2 - 14a + 48 = 0 \]

### Step 10: Factor the quadratic equation
Now, we can factor the quadratic:
\[ (a - 6)(a - 8) = 0 \]
So, \( a = 6 \) or \( a = 8 \).

### Step 11: Find the corresponding values of \( b \)
If \( a = 6 \), then:
\[ b = 14 - 6 = 8 \]
If \( a = 8 \), then:
\[ b = 14 - 8 = 6 \]
Thus, the sides of the triangle are \( 6 \) cm and \( 8 \) cm.

### Step 12: Calculate the area of the triangle
The area \( A \) of a right-angled triangle is given by:
\[ A = \frac{1}{2} \times a \times b \]
Substituting the values:
\[ A = \frac{1}{2} \times 6 \times 8 = \frac{48}{2} = 24 \text{ cm}^2 \]

### Conclusion
The area of the triangle is \( 24 \text{ cm}^2 \).

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