Your question: "The shorter diagonal of a rhombus is 50 mm long. Each of its obtuse angles measures 140. Find the length of each side."

Then each of the acute angles is 40, and the the angle bisected by the shorter diagonol divides into two 70deg

law of sines:
50mm/Sin40= s/Sne70

To find the length of each side of the rhombus, we can use the properties of a rhombus.

In a rhombus, the diagonals bisect each other at right angles, and the diagonals are not equal in length.

Given that the shorter diagonal is 50 mm long and each of the obtuse angles measures 140 degrees, we can solve for the length of each side using trigonometry.

We can divide the rhombus into two congruent right-angled triangles using the shorter diagonal.

Let's label the length of each side of the rhombus as "s" (which is what we are trying to find).

In each triangle, the hypotenuse is the side of the rhombus, and the shorter diagonal is the base of the triangle.

Using the trigonometric property cosine (cos), we can write the equation:
cos(140°) = adjacent/hypotenuse

Since the adjacent side is half of the length of the rhombus side (s/2), and the hypotenuse is the length of the shorter diagonal (50 mm), we can rewrite the equation as:
cos(140°) = (s/2)/50

Next, we solve for s:
Multiply both sides of the equation by 50 to get rid of the denominator:
50 * cos(140°) = (s/2)
s = 100 * cos(140°)

Using a calculator, evaluate cos(140°):
cos(140°) ≈ -0.766

Substitute this value back into the equation to find s:
s = 100 * (-0.766)
s ≈ -76.6

Since lengths cannot be negative, the length of each side of the rhombus is approximately 76.6 mm.

To find the length of each side of the rhombus, we can use the properties of a rhombus.

A rhombus is a quadrilateral with all sides of the same length. It also has opposite angles that are congruent. In this case, we know that one of the angles is obtuse and measures 140 degrees.

To find the length of each side, we need to first find the length of the longer diagonal. Since the diagonals of a rhombus are perpendicular, they will create right angles with each other.

The diagonals of a rhombus bisect each other, which means that they divide each other into two equal segments. Therefore, the longer diagonal will have a length equal to twice the length of the shorter diagonal.

We are given that the shorter diagonal is 50 mm long. Therefore, the longer diagonal will be 2 times 50, which is 100 mm.

Now, let's use the diagonals of the rhombus to find the length of each side.

The diagonals of a rhombus divide it into four congruent right triangles. We can use one of these triangles to find the length of a side.

In the right triangle formed by the shorter diagonal and half of the longer diagonal, we know the measure of one of the acute angles (90 degrees) and the measure of one of the obtuse angles (140 degrees). We can use these angles to find the measure of the remaining acute angle.

The sum of the three angles in a triangle is always 180 degrees. Therefore, we can subtract the measures of the two known angles from 180 to find the measure of the third angle.

180 - 140 - 90 = 50 degrees.

Now we have an acute angle of 50 degrees in the right triangle. We can use trigonometry to find the length of the side.

In the right triangle, the length of the shorter side (the side opposite the acute angle) is equal to half the length of the longer diagonal (50 mm). The length of the hypotenuse (the side adjacent to the right angle) is the length of a side of the rhombus (which we want to find).

Using the sine function (sin), we can set up the following equation:

sin(50 degrees) = shorter side / hypotenuse

sin(50 degrees) = 50 mm / side length

To solve for the side length, we can multiply both sides of the equation by the side length and divide by sin(50 degrees):

side length = (50 mm) / sin(50 degrees)

Using a calculator, we can find that sin(50 degrees) is approximately 0.766.

Therefore, the length of each side of the rhombus is:

side length = (50 mm) / 0.766

side length ≈ 65.22 mm (rounded to two decimal places)

So, the length of each side of the rhombus is approximately 65.22 mm.