Equilateral Triangle Properties and Altitude BD Calculation

The question asks us to find the value of \(BD^2\) in relation to \(AD^2\), given that ABC is an equilateral triangle and BD is drawn perpendicular to AC.

Properties of Equilateral Triangle ABC

Let's recall the fundamental properties of an equilateral triangle:

• All sides are equal in length. Let's denote the side length as \(s\). So, \(AB = BC = AC = s\).
• All interior angles are equal to \(60^\circ\). Hence, \(\angle A = \angle B = \angle C = 60^\circ\).

Altitude BD Characteristics in Equilateral Triangle

When a line segment BD is drawn from vertex B perpendicular to the opposite side AC:

• BD is the altitude to side AC. This means \(\angle BDA = 90^\circ\).
• A key property of altitudes in equilateral triangles is that they also bisect the base. So, the point D is the midpoint of AC.
• This implies that the length of AD is exactly half the length of AC. Mathematically, \(AD = \frac{AC}{2}\).
• Since \(AC = s\), we have \(AD = \frac{s}{2}\).
• We can also relate the side length AB to AD. Since \(AB = s\) and \(AD = \frac{s}{2}\), it follows that \(AB = 2 \times AD\).

Pythagorean Theorem Application for \(BD^2\)

Now, let's focus on the right-angled triangle formed, \(\triangle ADB\). We know \(\angle ADB = 90^\circ\). We can apply the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the \(90^\circ\) angle) is equal to the sum of the squares of the other two sides.

In \(\triangle ADB\):

• Hypotenuse = \(AB\)
• Other two sides = \(AD\) and \(BD\)

Applying the theorem, we get:

\(BD^2\) Derivation Using \(AD^2\)

We need to find what \(BD^2\) is equal to in terms of \(AD^2\). We established earlier that \(AB = 2 \times AD\) for our equilateral triangle scenario.

Let's substitute this relationship into the Pythagorean equation:

1. Start with: \(AB^2 = AD^2 + BD^2\)
2. Substitute \(AB = 2 \times AD\): \( (2 \times AD)^2 = AD^2 + BD^2 \)
3. Simplify the left side: \( 4 \times AD^2 = AD^2 + BD^2 \)
4. Now, rearrange the equation to solve for \(BD^2\). Subtract \(AD^2\) from both sides: \( BD^2 = 4 \times AD^2 - AD^2 \)
5. Perform the subtraction: \( BD^2 = 3 \times AD^2 \)

Final Result for \(BD^2\)

Based on the derivation using the properties of an equilateral triangle and the Pythagorean theorem, we find that \(BD^2\) is equal to 3 times \(AD^2\).