Altitude and base of a right angle triangle are (x+2) and (2x+3) ( in cm ). If the area of the triangle be 60 ` cm^(2)` , the length of the hypotenuse is :

Options

A. 21 cm
B. 13 cm
C. 17 cm
D. 15 cm

Answer

To find the length of the hypotenuse of a right-angled triangle with given altitude and base, we can follow these steps:

### Step 1: Write down the formula for the area of a triangle.
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Substitute the given values into the area formula.
Here, the altitude (height) is \( (x + 2) \) cm and the base is \( (2x + 3) \) cm. The area is given as 60 cm². Thus, we can set up the equation:
\[ \frac{1}{2} \times (2x + 3) \times (x + 2) = 60 \]

### Step 3: Simplify the equation.
Multiply both sides by 2 to eliminate the fraction:
\[ (2x + 3)(x + 2) = 120 \]

### Step 4: Expand the left side of the equation.
Using the distributive property (FOIL method):
\[ 2x^2 + 4x + 3x + 6 = 120 \]
Combine like terms:
\[ 2x^2 + 7x + 6 = 120 \]

### Step 5: Move all terms to one side to form a quadratic equation.
Subtract 120 from both sides:
\[ 2x^2 + 7x + 6 - 120 = 0 \]
This simplifies to:
\[ 2x^2 + 7x - 114 = 0 \]

### Step 6: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 7 \), and \( c = -114 \). First, calculate the discriminant:
\[ b^2 - 4ac = 7^2 - 4 \times 2 \times (-114) = 49 + 912 = 961 \]
Now, substitute into the quadratic formula:
\[ x = \frac{-7 \pm \sqrt{961}}{2 \times 2} \]
Since \( \sqrt{961} = 31 \):
\[ x = \frac{-7 \pm 31}{4} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{24}{4} = 6 \)
2. \( x = \frac{-38}{4} \) (not valid since \( x \) must be positive)

Thus, \( x = 6 \).

### Step 7: Calculate the base and height.
Substituting \( x = 6 \) back into the expressions for base and height:
- Height: \( x + 2 = 6 + 2 = 8 \) cm
- Base: \( 2x + 3 = 2(6) + 3 = 12 + 3 = 15 \) cm

### Step 8: Use the Pythagorean theorem to find the hypotenuse.
The Pythagorean theorem states:
\[ \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \]
Substituting the values:
\[ \text{hypotenuse}^2 = 15^2 + 8^2 = 225 + 64 = 289 \]
Taking the square root:
\[ \text{hypotenuse} = \sqrt{289} = 17 \text{ cm} \]

### Final Answer:
The length of the hypotenuse is \( 17 \) cm.

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