math destroyer rhombus question

[spoog74]

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question asks; if two sides of a rhombus meet at an angle of 60 degrees and one of the sides is of the length 20, what is the area of the rhombus? The answer is given 200 square root of 3.

The explanation says that the hypotenuse is of length 20 given. <<--- how do they assume that the hypotenuse is 20? That is my confusion.....

 

[FeralisExtremum]

The reason they talk about a hypotenuse is because you are supposed to drop a vertical line down to form a triangle like they do here:

And use the given hypotenuse of that triangle (the length of the side, 20) to find the height, which can be used to calculate the area. Since you know it's a rhombus, and you're given the angle of 60 degrees, you use the relationship of a 30-60-90 triangle, which have a side ratio of 1: sqr(3) : 2, shown here:

Now, since you know that the hypotenuse is 20, the vertical height must be 10 sqr (3).

And a rhombus's area is base * height, since we know a rhombus has 4 sides of equal length, the base * height is:

20 * 10 * sqr (3) = 200 sqr (3)

Hope this helps.

 

[spoog74]

The reason they talk about a hypotenuse is because you are supposed to drop a vertical line down to form a triangle like they do here:

And use the given hypotenuse of that triangle (the length of the side, 20) to find the height, which can be used to calculate the area. Since you know it's a rhombus, and you're given the angle of 60 degrees, you use the relationship of a 30-60-90 triangle, which have a side ratio of 1: sqr(3) : 2, shown here:

Now, since you know that the hypotenuse is 20, the vertical height must be 10 sqr (3).

And a rhombus's area is base * height, since we know a rhombus has 4 sides of equal length, the base * height is:

20 * 10 * sqr (3) = 200 sqr (3)

Hope this helps.

maybe i just dont understand, but who is to say the the hypotenuse gets the length of 20 and not the base of the whole figure? Why do we assume the hyp is 20?

 

[FeralisExtremum]

The base of the figure IS also 20 - remember, it's a rhombus, all side lengths are equal. So when they tell you a side length is 20, you know that the slanted side length must be 20. That slanted side length is what forms the hypotenuse of the right triangle we temporarily create to determine the height. The slanted side length HAS to be the hypotenuse because it will always be longer than both the vertical line and the horizontal triangle base (because this is only a portion of the rhombus's base, which we already know is the same length as the hypotenuse).

Look again at the first image in my first post to see this illustrated better - the base of that triangle HAS to be less than the hypotenuse, because s is the hypotenuse and the base of that triangle must always be less than the full length of s.

 

[dl2501]

@orgoman22

Dr. Romano,

For this question, if the rhombus is basically a slanted square, why isn't the area just the base times height...S^2. In this case the the area would be 20^2=400. For a square, the area is just base times height, and a rhombus is just a slanted square, and no area should be displaced. It seems like I may be missing something here....

Thank you!

 

[DAT Destroyer]

@orgoman22

Dr. Romano,

For this question, if the rhombus is basically a slanted square, why isn't the area just the base times height...S^2. In this case the the area would be 20^2=400. For a square, the area is just base times height, and a rhombus is just a slanted square, and no area should be displaced. It seems like I may be missing something here....

Thank you!

If your thinking is right then the area of a parallelogram should be the same as the area of a rectangle, which is not the case.

The areas of a rectangle, a square, a rhombus, and a parallelogram are given by: Area = Base* height. The height is the perpendicular distance from the base to the opposite side. In the case of a square and a rectangle, the height is just the other side since it's perpendicular to the base.
In the case of a rhombus and a parallelogram, the height is not the other side