Given: ABCD is a parallelogram
m∠A = 60º ; BK ⊥AD AK = KD; Perimeter of ABCD = 24
Find: BD.
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To find BD, we need to analyze the given information and solve step by step. Here's how you can determine the length of BD:
Step 1: Recognize properties of a parallelogram:
Since ABCD is a parallelogram, opposite sides are equal in length. Therefore, AD = BC and AB = CD.
Step 2: Determine triangle angles:
Given that m∠A = 60º, it can be concluded that m∠D = 180º - 60º = 120º because opposite angles in a parallelogram are congruent. Similarly, m∠B = 120º.
Step 3: Analyze triangle ABK:
BK ⊥ AD implies angle AKB is 90º. Since AK = KD, triangle ABK is an isosceles triangle. Therefore, m∠BAK = m∠BKA.
Step 4: Apply triangle perimeter information:
The perimeter of ABCD is given as 24 units. Perimeter equals the sum of all four sides: AB + BC + CD + AD. As AB = CD and AD = BC, 2AB + 2AD = 24. Simplifying, we have AB + AD = 12.
Step 5: Apply triangle angle sum:
In triangle ABK, m∠BAK + m∠BKA + m∠AKB = 180º. Since m∠BAK = m∠BKA, this becomes 2∠BAK + m∠AKB = 180º.
Step 6: Set up equations and solve:
From Step 4, AB + AD = 12. From Step 5, 2∠BAK + m∠AKB = 180º. As ∠BAK and ∠AKB are equal, let's assign x to represent this angle. This results in x + x + 120º = 180º. Simplifying, we get 2x + 120º = 180º, which leads to 2x = 60º, and eventually, x = 30º.
Step 7: Determine lengths using trigonometry:
In triangle ABK, we can use trigonometry to find the lengths of AB and AK. Since ∠BAK = ∠AKB = 30º, we can use the 30-60-90 triangle ratios. In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2.
Therefore, AB/1 = AK/(√3) = AK/2. Since AB + AK = 12 (from Step 4), we can write the equation AB + AB√3 = 12. Solving for AB, we have AB = 12/(1 + √3).
Next, we can find AK by using AK = 2AB/(1 + √3).
Finally, we can find BD by using BD = AK + KD. Since AK = KD, we have BD = 2AK.
Now that you have the methodology, you can calculate the value of BD using these steps.
Sure, I can help you with that! Let's go step-by-step to find the length of BD.
Step 1: Draw the diagram
Start by sketching the parallelogram ABCD with the given information. Label the angles and sides as mentioned:
A___________B
/ \
/ \
/ \
D________________C
Given:
m∠A = 60º
BK ⊥ AD
AK = KD
Perimeter of ABCD = 24
Step 2: Use the given information to solve the problem
We have to find the length of BD.
Since ABCD is a parallelogram, we know that opposite sides are equal in length.
We are given that the perimeter of ABCD is 24. The perimeter of a parallelogram is calculated by adding all four sides.
Let's assume side AB = CD = x and side BC = AD = y.
Then, the perimeter equation becomes:
AB + BC + CD + AD = 2x + 2y = 24
Simplifying the equation:
2(x + y) = 24
x + y = 12 ----(1)
Step 3: Use the properties of a parallelogram to find the length of BD
In a parallelogram, opposite angles are equal. Given that m∠A = 60º, we know that m∠C is also 60º.
Let's consider ΔAOC. It is an equilateral triangle since it has three angles measuring 60º each.
By symmetry, we can conclude that ∠BOC, ∠COD, and ∠DOC are also 60º each.
In ΔCOD, we know ∠COD = ∠C = 60º and ∠CDO = 90º (since BK ⊥ AD).
Using the properties of triangles, the sum of the angles in ΔCOD is 180º. Therefore, ∠ODC = 180º - 90º - 60º = 30º.
Step 4: Use trigonometric ratios to find the length of BD
In ΔODB, we have an angle of 30º and a side length of x, which is half of BD.
Looking at the trigonometric ratios for a 30º angle in a right-angled triangle, we have:
sin(30º) = opposite / hypotenuse
sin(30º) = x / BD/2
We know that sin(30º) = 1/2, so we can rewrite the equation as:
1/2 = x / BD/2
Cross-multiplying the equation, we get:
BD = 2 * x / (1/2)
BD = 4x
Step 5: Plug in the value of x from equation (1)
From equation (1), we know that x + y = 12.
Since AB = CD = x, substituting x = 12 - y into the equation for BD, we get:
BD = 4 * (12 - y)
Step 6: Solve for BD
To find the length of BD, we need to know the value of y.
Since AK = KD, we can conclude that AD is divided into two equal parts.
Therefore, y = (AD/2) = (2x) / 2 = x.
Substituting x = y in the equation for BD, we get:
BD = 4 * (12 - y)
BD = 4 * (12 - x)
BD = 4 * (12 - y)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
BD = 4 * (12 - x)
Draw an altitude from D to AB. It has length h.
Let AB = x, AK=KD=y, BD=z
since m∠A = 60º, h=y√3
Now you have 4 equations in x,y,z,h:
2x+4y=24
x^2 = y^2+z^2
hx=2yz (area=base*height)
h=√3 y
Solve those and you will see that the parallelogram is a rhombus of side 6, with both altitudes 3√3