Question
Given:ABCD Is a parallelogram, m<A=60 degrees, line BK is perpendicular to line AD, AK = KD,
Permimeter of ABCD = 24.
Find:BD
Permimeter of ABCD = 24.
Find:BD
Answers
GPT-4o mini
To find the length of diagonal \(BD\) in parallelogram \(ABCD\) with given conditions, we'll follow these steps:
1. **Understanding the given information:**
- \(m \angle A = 60^\circ\)
- \(BK \perp AD\) (line \(BK\) is perpendicular to line \(AD\))
- \(AK = KD\) (which means \(K\) is the midpoint of \(AD\))
- The perimeter of parallelogram \(ABCD\) is 24.
2. **Calculating the side lengths:**
Let the lengths of sides \(AB\) and \(AD\) be \(a\) and \(b\) respectively.
Since the perimeter \(P = 2(a + b)\), we have:
\[
2(a + b) = 24 \implies a + b = 12.
\]
3. **Using properties of the parallelogram:**
In a parallelogram, opposite sides are equal:
\[
AB = CD = a \quad \text{and} \quad AD = BC = b.
\]
Thus, we can express one side in terms of the other:
\[
b = 12 - a.
\]
4. **Applying the trigonometric properties:**
Let’s break down triangle \(AKD\):
- Since \(AK = KD = \frac{b}{2}\) (since \(K\) is the midpoint).
- The height \(BK\) from \(B\) to line \(AD\) can be expressed in terms of \(b\) and the angle \(\angle A\).
\[
BK = AB \sin A = a \sin 60^\circ = a \cdot \frac{\sqrt{3}}{2}.
\]
5. **Finding the length \(BD\):**
We can use the formula for the diagonal \(BD\):
\[
BD^2 = AB^2 + AD^2 - 2(AB)(AD) \cos A,
\]
where \(\cos(60^\circ) = \frac{1}{2}\). Substituting what we have:
\[
BD^2 = a^2 + b^2 - 2ab \cdot \frac{1}{2} = a^2 + b^2 - ab.
\]
6. **Substituting \(b = 12 - a\):**
\[
BD^2 = a^2 + (12 - a)^2 - a(12 - a),
\]
\[
= a^2 + (144 - 24a + a^2) - (12a - a^2) = 2a^2 - 12a + 144.
\]
7. **Finding the optimal values:**
To find out \( BD \) for different values of \( a \) satisfying \( a + b = 12 \), we are basically looking for optimal side lengths. Therefore we can evaluate when:
- Use \( a = b = 6 \) (maximizing symmetry):
\[
BD = \sqrt{2a^2 - 12a + 144} = \sqrt{2(6^2) - 12(6) + 144} = \sqrt{72 - 72 + 144} = \sqrt{144} = 12.
\]
Thus, the length of diagonal \(BD\) is **12** units.
1. **Understanding the given information:**
- \(m \angle A = 60^\circ\)
- \(BK \perp AD\) (line \(BK\) is perpendicular to line \(AD\))
- \(AK = KD\) (which means \(K\) is the midpoint of \(AD\))
- The perimeter of parallelogram \(ABCD\) is 24.
2. **Calculating the side lengths:**
Let the lengths of sides \(AB\) and \(AD\) be \(a\) and \(b\) respectively.
Since the perimeter \(P = 2(a + b)\), we have:
\[
2(a + b) = 24 \implies a + b = 12.
\]
3. **Using properties of the parallelogram:**
In a parallelogram, opposite sides are equal:
\[
AB = CD = a \quad \text{and} \quad AD = BC = b.
\]
Thus, we can express one side in terms of the other:
\[
b = 12 - a.
\]
4. **Applying the trigonometric properties:**
Let’s break down triangle \(AKD\):
- Since \(AK = KD = \frac{b}{2}\) (since \(K\) is the midpoint).
- The height \(BK\) from \(B\) to line \(AD\) can be expressed in terms of \(b\) and the angle \(\angle A\).
\[
BK = AB \sin A = a \sin 60^\circ = a \cdot \frac{\sqrt{3}}{2}.
\]
5. **Finding the length \(BD\):**
We can use the formula for the diagonal \(BD\):
\[
BD^2 = AB^2 + AD^2 - 2(AB)(AD) \cos A,
\]
where \(\cos(60^\circ) = \frac{1}{2}\). Substituting what we have:
\[
BD^2 = a^2 + b^2 - 2ab \cdot \frac{1}{2} = a^2 + b^2 - ab.
\]
6. **Substituting \(b = 12 - a\):**
\[
BD^2 = a^2 + (12 - a)^2 - a(12 - a),
\]
\[
= a^2 + (144 - 24a + a^2) - (12a - a^2) = 2a^2 - 12a + 144.
\]
7. **Finding the optimal values:**
To find out \( BD \) for different values of \( a \) satisfying \( a + b = 12 \), we are basically looking for optimal side lengths. Therefore we can evaluate when:
- Use \( a = b = 6 \) (maximizing symmetry):
\[
BD = \sqrt{2a^2 - 12a + 144} = \sqrt{2(6^2) - 12(6) + 144} = \sqrt{72 - 72 + 144} = \sqrt{144} = 12.
\]
Thus, the length of diagonal \(BD\) is **12** units.