Rhombus help please.?

Question

The area of a rhombus is 300 square inches. The diagonals are in the ratio of 2:3. Find the length of a side of the rhombus.

Answer 1

The area of a rhombus is half the product of the diagonals. Let's call the diagonals 2x and 3x, so...

d1 * d2 / 2 = area
(2x)*(3x) / 2 = 300
3x^2 = 300
x = 10

You know that the diagonals are 20 (2x) and 30 (3x) inches.

If you take half of each diagonal (10 and 15), you have the legs of a right triangle, where the hypotenuse is one of the sides of the rhombus. Next, use the pythagorean theorem:

a^2 + b^2 = c^2
10^2 + 15^2 = c^2
100 + 225 = c^2
325 = c^2
sqrt(325) = c
sqrt(25 * 13) = c
5 * sqrt(13) = c

The length of a side is 5 * sqrt(13) inches, which is about 18 inches. (It's 18.027756)

Answer 2

Since the diagonals of a rhombus always meet at a right angle, each side must be the hypotenuse of a right triangle formed by half-diagonals. The first step is to determine the angles of this right triangle.

Since the two half-diagonals are also in a 2:3 ratio, the hypotenuse (rhombus side) must be sqrt(4+9) = sqrt(13) = 3.61. The smaller acute angle (of the right-triangle) must then be arctan(3 / 3.61) = 39.8°.

Since each acute interior angle of the rhombus is composed of two of these (do you see why? If not, make a drawing), the acute interior angle of the rhombus must be 79.5°.

The area of a rhombus is base x height, and in this case b x h = 300. We also know that the height must be sin (79.5) * b. So the area equation becomes
b * [sin (79.5) * b] = 300
or
sin (79.5) * b^2 = 300
or
.98333 b^2 = 300

dividing by .98333 we get
b^2 = 305.09
and
b = 17.4667
... which is the length of one side.

Answer 3

your answer should be 275 sq. inches