The perimeter of a right triangle is 90 cm and its area is ` 270 cm ^(2)` . Find the hypotenuse

Options

A. 39 cm
B. 41 cm
C. 52 cm
D. 36 cm

Answer

To find the hypotenuse of a right triangle given its perimeter and area, we can follow these steps:

### Step 1: Define the variables
Let the lengths of the two legs of the right triangle be \( a \) and \( b \), and the hypotenuse be \( c \).

### Step 2: Set up the equations
From the problem, we have two equations based on the perimeter and area:

1. The perimeter of the triangle:
\[ a + b + c = 90 \quad \text{(1)} \]

2. The area of the triangle:
\[ \frac{1}{2}ab = 270 \quad \Rightarrow \quad ab = 540 \quad \text{(2)} \]

### Step 3: Express \( c \) in terms of \( a \) and \( b \)
From equation (1), we can express \( c \):
\[ c = 90 - a - b \quad \text{(3)} \]

### Step 4: Use the Pythagorean theorem
For a right triangle, we have:
\[ c^2 = a^2 + b^2 \quad \text{(4)} \]

### Step 5: Substitute \( c \) from equation (3) into equation (4)
Substituting equation (3) into equation (4):
\[ (90 - a - b)^2 = a^2 + b^2 \]

### Step 6: Expand and simplify
Expanding the left side:
\[ 8100 - 180a - 180b + a^2 + 2ab + b^2 = a^2 + b^2 \]
Cancelling \( a^2 + b^2 \) from both sides gives:
\[ 8100 - 180a - 180b + 2ab = 0 \]
Rearranging this, we get:
\[ 2ab - 180a - 180b + 8100 = 0 \]

### Step 7: Substitute \( ab \) from equation (2)
Substituting \( ab = 540 \) into the equation:
\[ 2(540) - 180a - 180b + 8100 = 0 \]
This simplifies to:
\[ 1080 - 180a - 180b + 8100 = 0 \]
\[ -180a - 180b + 9180 = 0 \]
Dividing the entire equation by -180:
\[ a + b = 51 \quad \text{(5)} \]

### Step 8: Solve for \( a \) and \( b \)
Now we have two equations:
1. \( ab = 540 \) (from equation (2))
2. \( a + b = 51 \) (from equation (5))

We can express \( b \) in terms of \( a \):
\[ b = 51 - a \]
Substituting this into \( ab = 540 \):
\[ a(51 - a) = 540 \]
This expands to:
\[ 51a - a^2 = 540 \]
Rearranging gives:
\[ a^2 - 51a + 540 = 0 \]

### Step 9: Solve the quadratic equation
Using the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = \frac{51 \pm \sqrt{51^2 - 4 \cdot 1 \cdot 540}}{2 \cdot 1} \]
Calculating the discriminant:
\[ 51^2 - 2160 = 2601 - 2160 = 441 \]
Now substituting back:
\[ a = \frac{51 \pm 21}{2} \]
This gives us two possible values for \( a \):
1. \( a = \frac{72}{2} = 36 \)
2. \( a = \frac{30}{2} = 15 \)

### Step 10: Find \( b \)
Using \( b = 51 - a \):
- If \( a = 36 \), then \( b = 51 - 36 = 15 \).
- If \( a = 15 \), then \( b = 51 - 15 = 36 \).

### Step 11: Calculate the hypotenuse \( c \)
Using equation (3):
\[ c = 90 - a - b = 90 - 36 - 15 = 39 \quad \text{(or the same for the other case)} \]

### Final Answer
The hypotenuse \( c \) is \( 39 \) cm.

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