In a ∆ABC, AD _|_BC and AD² = BD X CD. Prove that A ABC is a right- angled triangle.

To prove that triangle \( \Delta ABC \) is a right-angled triangle given that \( AD \perp BC \) (i.e., \( AD \) is perpendicular to \( BC \)) and \( AD^2 = BD \times CD \), we can proceed as follows:

### Given:
1. \( \Delta ABC \) with \( AD \perp BC \).
2. \( AD^2 = BD \times CD \).

### To Prove:
\( \Delta ABC \) is a right-angled triangle.

### Proof:

1. Let the lengths of the segments be as follows:
   - \( BD = x \)
   - \( CD = y \)
   - \( AD = h \)

   So, the given condition is:
   \[
   h^2 = x \times y
   \]

2. In \( \Delta ABD \) and \( \Delta ACD \), using the fact that \( AD \) is the altitude to the hypotenuse in right triangles:
   - By the **Altitude-on-Hypotenuse Theorem** (or Geometric Mean Theorem), we know:
   \[
   AB^2 = BD \times BC
   \]
   \[
   AC^2 = CD \times BC
   \]
   Since \( BC = BD + CD = x + y \), we have:
   \[
   AB^2 = x(x + y)
   \]
   \[
   AC^2 = y(x + y)
   \]

3. Also, the area of \( \Delta ABC \) can be expressed in two ways:
   - Using the base \( BC \) and height \( AD \):
   \[
   \text{Area} = \frac{1}{2} \times BC \times AD = \frac{1}{2} \times (x + y) \times h
   \]
   - Using Heron's formula or other combinations:
   \[
   \text{Area} = \frac{1}{2} \times AB \times AC \times \sin \angle BAC
   \]

4. Since \( h^2 = x \times y \), let's rewrite \( h = \sqrt{x \times y} \).

   Substituting \( h = \sqrt{x \times y} \) into the area expression gives us:
   \[
   \text{Area} = \frac{1}{2} \times (x + y) \times \sqrt{x \times y}
   \]

5. The expression for \( AB^2 + AC^2 \) becomes:
   \[
   AB^2 + AC^2 = x(x + y) + y(x + y) = (x + y)(x + y) = (x + y)^2
   \]

6. Observe that:
   \[
   BC^2 = (x + y)^2
   \]

7. Since \( AB^2 + AC^2 = BC^2 \), by the **Pythagorean Theorem**, \( \Delta ABC \) is a right-angled triangle with the right angle at \( \angle BAC \).

### Conclusion:
\( \Delta ABC \) is indeed a right-angled triangle, with \( \angle BAC \) being the right angle.

Would you like any more details or have any questions?

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Here are five related questions:

1. How can the Geometric Mean Theorem be applied to other triangles?
2. What are the conditions under which a triangle has an altitude that satisfies \( AD^2 = BD \times CD \)?
3. How does the Pythagorean Theorem extend to non-right triangles?
4. What is the significance of the altitude in a right-angled triangle?
5. How do different methods of calculating a triangle's area compare?

**Tip:** Always check if a perpendicular segment in a triangle satisfies the relationship \( h^2 = BD \times CD \), as it often indicates the presence of a right angle.

Learn how to prove triangle ABC as a right-angled triangle given AD perpendicular to BC and AD^2 = BD * CD. This problem involves the application of geometric theorems like the Pythagorean theorem and the Altitude-on-Hypotenuse Theorem. Suitable for high school students studying geometry.

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