Each side of a rhombus is 14 in. long. Two of the sides form a 60-degree angle. Find the area of the rhombus. Round your answer to the nearest square inch.
I know how to solve special right triangles. I just don't know where to start. If someone could point me in a general direction to begin that would be great, thanks!
since the area is base * height,
pick a side as a base: 14
the height is thus 14 sin 60°
and you're home free.
To find the area of the rhombus, we need to first find the lengths of the diagonals.
Since two sides of the rhombus form a 60-degree angle, we can split the rhombus into two congruent 30-60-90 right triangles.
In a 30-60-90 right triangle, the lengths of the sides are related by the ratio:
opposite leg : adjacent leg : hypotenuse = x : x√3 : 2x
Let's call the length of the shorter side of the right triangle y. Therefore, the length of the longer side is y√3, and the hypotenuse is 2y.
In the rhombus, each side has a length of 14 inches, so the shorter side of the right triangle (y) is half of the side length, which is 7 inches.
Now we can use this information to find the length of the longer side (y√3) and the length of the hypotenuse (2y).
Substituting y = 7 into the ratios, we have:
Opposite leg = y = 7 inches
Adjacent leg = y√3 = 7√3 inches
Hypotenuse = 2y = 2(7) = 14 inches
So, we have found the lengths of the two diagonals of the rhombus: 7√3 inches and 14 inches.
Now we can use the formula for the area of a rhombus, which is given by:
Area = (diagonal1 * diagonal2) / 2
Substituting the lengths of the diagonals into the formula, we have:
Area = (7√3 * 14) / 2
= 98√3 / 2
= 49√3
Rounded to the nearest square inch, the area of the rhombus is approximately 84 square inches.