Respuesta :

sharmaenderp
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.

[tex] \overline{OB}:\overline{AB} = 1:2 [/tex]
[tex]\overline {OB}:10 = 1:2 [/tex]
[tex]\overline{OB} = \frac{1}{2}(10) = 5 [/tex]

Similarly, we have

[tex] \overline{AO}:\overline{AB} = \sqrt{3}:2 [/tex]
[tex]\overline {AO}:10 = \sqrt{3}:2 [/tex]
[tex]\overline{AO} = \frac{\sqrt{3}}{2}(10) = 5\sqrt{3} [/tex]

Now, to find the lengths of the diagonals, 

[tex]\overline{AD} = 2(\overline{AO}) = 10\sqrt{3} [/tex]
[tex]\overline{BC} = 2(\overline{OB}) = 10 [/tex]

So, the lengths of the diagonals are 10 and 10√3.

Answer: 10 and 10√3 units
Ver imagen sharmaenderp

Rhombus is a parallelogram but with equal side lengths. The lengths of the diagonals of ABCD rhombus are: 10 units and 10√3 units.

What is rhombus and some of its properties?

Rhombus is a parallelogram whose all sides are of equal lengths.

  • Its diagonals are perpendicular to each other and they cut each other in half( thus, they're perpendicular bisector of each other).
  • Its vertex angles are bisected by its diagonals.
  • The triangles on either side of the diagonals are isosceles and congruent.

What is Pythagoras theorem?

If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:

[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]

where, |AB| = length of AB line segment.

Since all sides of rhombus are equal, and given that |AB| = 10, thus, all sides are of 10 units length.

If we take the triangle AOC, we have angle O internal to triangle AOC as of 90 degrees as diagonals are perpendicular to each other.

Also, since they bisect each other, thus,

  • Length of diagonal AC = |AC| = 2|AO| = 2|OC|
  • Length of diagonal BD = |BD| = 2|BO| = 2|OD|

using the sin trigonometric ratio, seeing from angle OAC, we get:

[tex]\sin(\angle OAC) = \dfrac{|OB|}{|AC|}\\\\\sin(30^\circ) = \dfrac{|OB|}{10}\\\dfrac{10}{2} = |OB|\\\\|OB| = 5\: \rm units[/tex]

Thus, length of diagonal BD = 2|OB| = 10 units.

Similarly, using cos function, we get:

[tex]\cos(\angle OAC) = \dfrac{|AO|}{|AC|}\\\\\cos(30^\circ) = \dfrac{|AO|}{10}\\\dfrac{10\sqrt{3}}{2} = |OB|\\\\|AO| = 5\sqrt{3}\: \rm units[/tex]

Thus, length of diagonal AC = 2|AO| = 10√3 units.

Thus,

The lengths of the diagonals of ABCD rhombus are: 10 units and 10√3 units.

Learn more about trigonometric ratios here:

https://brainly.com/question/22599614

Ver imagen astha8579