In `Delta ABC, If AD _|_ BC,` then `AB^2 + CD^2` is equal to

Options

A. `2 BD^2`
B. `BD^2 + AC^2`
C. `2 AC^2`
D. None of these

Answer

To solve the problem, we will use the properties of right triangles and the Pythagorean theorem. Let's break down the solution step by step.

### Step 1: Understand the Triangle
We have triangle ABC where AD is perpendicular to BC. This means that triangles ABD and ACD are right triangles.

**Hint:** Visualize the triangle and mark the right angles to understand the relationships between the sides.

### Step 2: Apply the Pythagorean Theorem in Triangle ABD
In triangle ABD, we can apply the Pythagorean theorem:
\[ AB^2 = AD^2 + BD^2 \quad \text{(1)} \]

**Hint:** Recall that in a right triangle, the square of the hypotenuse (AB) is equal to the sum of the squares of the other two sides (AD and BD).

### Step 3: Apply the Pythagorean Theorem in Triangle ACD
Now, in triangle ACD, we again apply the Pythagorean theorem:
\[ AC^2 = AD^2 + CD^2 \quad \text{(2)} \]

**Hint:** Similarly, identify the hypotenuse and the two sides in triangle ACD to set up the equation.

### Step 4: Rearranging the Equations
From equation (1), we can express \(AD^2\):
\[ AD^2 = AB^2 - BD^2 \quad \text{(3)} \]

From equation (2), we can express \(AD^2\) as well:
\[ AD^2 = AC^2 - CD^2 \quad \text{(4)} \]

**Hint:** Both equations (3) and (4) represent \(AD^2\), so we can set them equal to each other.

### Step 5: Equate the Two Expressions for \(AD^2\)
Setting equations (3) and (4) equal to each other gives:
\[ AB^2 - BD^2 = AC^2 - CD^2 \]

**Hint:** This step combines the information from both triangles to find a relationship between the sides.

### Step 6: Rearranging the Equation
Rearranging the equation gives:
\[ AB^2 + CD^2 = AC^2 + BD^2 \]

**Hint:** Isolate \(AB^2 + CD^2\) to see the final relationship clearly.

### Conclusion
Thus, we find that:
\[ AB^2 + CD^2 = AC^2 + BD^2 \]

This means that the sum of the squares of sides AB and CD is equal to the sum of the squares of sides AC and BD.

**Final Answer:** The expression \(AB^2 + CD^2\) is equal to \(AC^2 + BD^2\).