A point D is taken from the side BC of a right-angled triangle ABC, where AB is hypotenuse. Then

Options

A. `AB^2 + CD^2 = BC^2 + AD^2`
B. `CD^2 + BD^2 = 2 AD^2`
C. `AB^2 + AC^2 = 2 AD^2`
D. `AB^2 = AD^2 + BD^2`

Answer

To solve the problem, we will use the Pythagorean theorem in both the larger triangle ABC and the smaller triangle ACD. Here are the steps:

### Step 1: Apply the Pythagorean Theorem to Triangle ABC
In triangle ABC, where AB is the hypotenuse, we can write the Pythagorean theorem as:
\[ AB^2 = AC^2 + BC^2 \]
This is our **Equation 1**.

### Step 2: Apply the Pythagorean Theorem to Triangle ACD
In triangle ACD, where AD is the hypotenuse, we can also apply the Pythagorean theorem:
\[ AD^2 = AC^2 + CD^2 \]
This is our **Equation 2**.

### Step 3: Express AC^2 from Equation 2
From Equation 2, we can isolate \( AC^2 \):
\[ AC^2 = AD^2 - CD^2 \]
This is our **Equation 3**.

### Step 4: Substitute AC^2 in Equation 1
Now, we will substitute the expression for \( AC^2 \) from Equation 3 into Equation 1:
\[ AB^2 = (AD^2 - CD^2) + BC^2 \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:
\[ AB^2 + CD^2 = AD^2 + BC^2 \]

### Conclusion
Thus, we have derived the relationship:
\[ AB^2 + CD^2 = AD^2 + BC^2 \]

This is the required result.

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