Answer:
6√3 cm.
Explanation:
The angle between the length of a rectangle and its diagonal is 30°. If the length of the rectangle is 6cm, find the length of the diagonal.
Let's call the width of the rectangle "w" and the diagonal "d". We know that the angle between the length and diagonal is 30 degrees, which means that the angle between the width and diagonal is 90 - 30 = 60 degrees (since the angles in a triangle add up to 180 degrees).
Now, we can use trigonometry to find the length of the diagonal. Specifically, we can use the sine function, which relates the lengths of the sides of a right triangle to the angles opposite them:
sin(angle) = opposite/hypotenuse
In this case, the angle we're interested in is 60 degrees, the opposite side is the width "w", and the hypotenuse is the diagonal "d". So we can write:
sin(60) = w/d
Solving for "d", we get:
d = w/sin(60)
Now we just need to find the width of the rectangle. Since we know the length of the rectangle is 6cm, we can use the fact that the opposite sides of a rectangle are equal to each other to write:
w = length*sin(30) = 6sin(30)
We can simplify sin(30) to 1/2, so:
w = 6/2 = 3
Now we can substitute this value of "w" into our expression for "d" to get:
d = w/sin(60) = 3/sin(60)
We can simplify sin(60) to √3/2, so:
d = 3/(√3/2) = 6/√3 = 2√3 * 2
So the length of the diagonal is 2√3 times the length of the width, or approximately 3.46 times the length of the width. Therefore, the length of the diagonal is:
d = 2√3 * 3 = 6√3 cm.