In a right triangle, the square of hypotenuse is equal to twice the product of remaining two sides. If the base of this triangle is 12 cm, then the perpendicular of triangle will be

Options

A. 12 cm
B. 10/2 cm
C. 9 cm
D. 5 cm

Answer

To solve the problem step by step, we will follow the given information and apply the Pythagorean theorem along with the condition provided in the question.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the base of the triangle be \( B = 12 \) cm.
- Let the perpendicular be \( P \) cm.
- Let the hypotenuse be \( H \) cm.

2. **Use the Given Condition**:
- According to the problem, the square of the hypotenuse is equal to twice the product of the remaining two sides:
\[ H^2 = 2 \times B \times P \]
- Substituting the value of \( B \):
\[ H^2 = 2 \times 12 \times P = 24P \]

3. **Apply the Pythagorean Theorem**:
- The Pythagorean theorem states that:
\[ H^2 = B^2 + P^2 \]
- Substituting the value of \( B \):
\[ H^2 = 12^2 + P^2 = 144 + P^2 \]

4. **Set the Two Equations Equal**:
- Now we have two expressions for \( H^2 \):
\[ 24P = 144 + P^2 \]

5. **Rearrange the Equation**:
- Rearranging the equation gives:
\[ P^2 - 24P + 144 = 0 \]

6. **Factor the Quadratic Equation**:
- We need to factor the quadratic equation:
\[ (P - 12)(P - 12) = 0 \]
- This simplifies to:
\[ P - 12 = 0 \]

7. **Solve for \( P \)**:
- Thus, we find:
\[ P = 12 \text{ cm} \]

### Final Answer:
The perpendicular of the triangle is \( 12 \) cm.

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