The perimeter of a right angled triangle is 60 cm. Its hypotenuse is 26 cm. The area of the triangle is

Options

A. 120 `cm^(2)`
B. 240 `cm^(2)`
C. 390 `cm^(2)`
D. 780 `cm^(2)`

Answer

To solve the problem step by step, we will find the area of the right-angled triangle given its perimeter and hypotenuse.

### Step 1: Understand the given information
We know:
- The perimeter of the triangle \( P = 60 \) cm
- The hypotenuse \( H = 26 \) 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 denote:
- Base = \( b \)
- Height = \( h \)

Thus, we can write the equation for the perimeter:
\[ b + h + H = 60 \]
Substituting the value of \( H \):
\[ b + h + 26 = 60 \]
Now, simplify to find \( b + h \):
\[ b + h = 60 - 26 = 34 \quad \text{(Equation 1)} \]

### Step 3: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ H^2 = b^2 + h^2 \]
Substituting \( H = 26 \):
\[ 26^2 = b^2 + h^2 \]
Calculating \( 26^2 \):
\[ 676 = b^2 + h^2 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
We have two equations:
1. \( b + h = 34 \) (Equation 1)
2. \( b^2 + h^2 = 676 \) (Equation 2)

From Equation 1, we can express \( h \) in terms of \( b \):
\[ h = 34 - b \]

Now, substitute \( h \) in Equation 2:
\[ b^2 + (34 - b)^2 = 676 \]
Expanding \( (34 - b)^2 \):
\[ b^2 + (34^2 - 68b + b^2) = 676 \]
Calculating \( 34^2 \):
\[ b^2 + (1156 - 68b + b^2) = 676 \]
Combining like terms:
\[ 2b^2 - 68b + 1156 = 676 \]
Simplifying:
\[ 2b^2 - 68b + 480 = 0 \]
Dividing the entire equation by 2:
\[ b^2 - 34b + 240 = 0 \]

### Step 5: Factor the quadratic equation
To factor the equation \( b^2 - 34b + 240 = 0 \), we look for two numbers that multiply to \( 240 \) and add up to \( 34 \). The factors are \( 24 \) and \( 10 \):
\[ (b - 24)(b - 10) = 0 \]
Thus, \( b = 24 \) or \( b = 10 \).

### Step 6: Find corresponding height \( h \)
Using \( b = 24 \):
\[ h = 34 - 24 = 10 \]
Using \( b = 10 \):
\[ h = 34 - 10 = 24 \]
So, the base and height are \( 24 \) cm and \( 10 \) cm respectively.

### Step 7: Calculate the area of the triangle
The area \( A \) of a right-angled triangle is given by:
\[ A = \frac{1}{2} \times b \times h \]
Substituting the values:
\[ A = \frac{1}{2} \times 24 \times 10 = 120 \, \text{cm}^2 \]

### Final Answer
The area of the triangle is \( 120 \, \text{cm}^2 \).

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