A rhombus ABCD has AB = 10 and m∠A = 60°. Find the lengths of the diagonals of ABCD.
Its Actually 10 and 10√3
Three important properties of the diagonals of a rhombus that we need for this problem are:
1. the diagonals of a rhombus bisect each other
2. the diagonals form two perpendicular lines
3. the diagonals bisect the angles of the rhombus
First, we can let O be the point where the two diagonals intersect (as shown in the attached image). Using the properties listed above, we can conclude that ∠AOB is equal to 90° and ∠BAO = 60/2 = 30°.
Since a triangle's interior angles have a sum of 180°, then we have ∠ABO = 180 - 90 - 30 = 60°. This shows that the ΔAOB is a 30-60-90 triangle.
For a 30-60-90 triangle, the ratio of the sides facing the corresponding anges is 1:√3:2. So, since we know that AB = 10, we can compute for the rest of the sides.
Similarly, we have
Now, to find the lengths of the diagonals,
So, the lengths of the diagonals are 10 and 10√3.
Answer: 10 and 10√3 units
Answer smart boy >:-(
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To find the lengths of the diagonals of rhombus ABCD, we need to apply the properties of a rhombus.
1. In a rhombus, all sides are congruent. Since AB = 10, all sides of the rhombus are also 10 units in length.
2. In a rhombus, opposite angles are congruent. Thus, ∠A = ∠C, and ∠B = ∠D.
3. The diagonals of a rhombus bisect each other at right angles. This means that the diagonals intersect at a 90-degree angle.
Now, let's proceed with finding the lengths of the diagonals.
Since ∠A = 60°, we know that ∠C (opposite angle) is also 60°.
Using the properties of a rhombus, we can apply the Law of Cosines to find the lengths of the diagonals.
Let AC represent the length of one diagonal, and BD represent the length of the other diagonal.
Using the Law of Cosines for triangle ABC, we can express AC in terms of the known sides:
AC^2 = AB^2 + BC^2 - 2 * AB * BC * cos(∠ABC)
Since the rhombus has congruent angles, we have:
AC^2 = 10^2 + 10^2 - 2 * 10 * 10 * cos(60°)
AC^2 = 100 + 100 - 200 * 0.5
AC^2 = 200 - 100
AC^2 = 100
Taking the square root of both sides, we find:
AC = sqrt(100)
AC = 10
So, one diagonal, AC, has a length of 10 units.
Since the diagonals of a rhombus bisect each other at right angles, we can use the Pythagorean Theorem to find the length of the other diagonal, BD.
BD^2 = AB^2 + AD^2
where AD is half of AC.
AD = AC / 2 = 10 / 2 = 5
BD^2 = 10^2 + 5^2
BD^2 = 100 + 25
BD^2 = 125
Taking the square root of both sides, we find:
BD = sqrt(125)
BD = 11.18 (approx.)
So, the lengths of the diagonals of rhombus ABCD are approximately 10 units and 11.18 units, respectively.
its 5 square root 7
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A rhombus has four sides equal in length and opposite sides parallel. Opposite interior angles are equal. Also, the diagonals bisect the interior angles.
Try constructing a rough sketch of a rhombus with a pencil and ruler. Label all 4 sides with"10" and two opposite angles 60 degrees (choose the two acute angles for this). You should be able to figure out the other two opposite angles. Draw in the diagonals, and you should have enough information to use the cosine law to find their lengths.