In the rhombus ABCD, angle C is 60°. In a rhombus, the opposite angles are equal. So, angle A is also 60°. Since the sum of angles in a quadrilateral is 360°, we can find angle B and angle D by subtracting the sum of angles A and C from 360°. Once we have all the angles, we can use trigonometric ratios to find the ratio of the sides AC and BD.
In a rhombus, opposite angles are congruent, and all angles are equal. Since you mentioned that angle C in the rhombus ABCD is 60 degrees, that means all angles in the rhombus are 60 degrees each.
Now, in a rhombus, the diagonals bisect each other at right angles, forming four right-angled triangles. The ratio of the lengths of the diagonals in a rhombus can be calculated using the Pythagorean theorem.
Let's denote AC as "a" and BD as "b." In each right-angled triangle, the hypotenuse (which is half of AC or BD) is opposite the 60-degree angle, and the other two sides are equal because the diagonals bisect each other.
Using the trigonometric relationship for a 30-60-90 triangle (opposite/hypotenuse = 1/2), you can set up the following equations:
For triangle ACB:
sin(60 degrees) = (1/2) * (AC/2) / (b/2)
Simplifying:
√3/2 = (1/2) * (a/2) / (b/2)
Now, canceling out common factors and simplifying further:
Answers & Comments
Answer:
In the rhombus ABCD, angle C is 60°. In a rhombus, the opposite angles are equal. So, angle A is also 60°. Since the sum of angles in a quadrilateral is 360°, we can find angle B and angle D by subtracting the sum of angles A and C from 360°. Once we have all the angles, we can use trigonometric ratios to find the ratio of the sides AC and BD.
Answer:
In a rhombus, opposite angles are congruent, and all angles are equal. Since you mentioned that angle C in the rhombus ABCD is 60 degrees, that means all angles in the rhombus are 60 degrees each.
Now, in a rhombus, the diagonals bisect each other at right angles, forming four right-angled triangles. The ratio of the lengths of the diagonals in a rhombus can be calculated using the Pythagorean theorem.
Let's denote AC as "a" and BD as "b." In each right-angled triangle, the hypotenuse (which is half of AC or BD) is opposite the 60-degree angle, and the other two sides are equal because the diagonals bisect each other.
Using the trigonometric relationship for a 30-60-90 triangle (opposite/hypotenuse = 1/2), you can set up the following equations:
For triangle ACB:
sin(60 degrees) = (1/2) * (AC/2) / (b/2)
Simplifying:
√3/2 = (1/2) * (a/2) / (b/2)
Now, canceling out common factors and simplifying further:
√3 = a / b
So, the ratio AC : BD is √3 : 1.