The lengths of 2 adjacent sides of a parallelogram are 42 cm and 36cm. an angle of the parallelogram is 4o degrees. Find the measure of the longest diagonal to the nearest tenth of a centimeter.

The longest diagonal forms a triangle with 180°-40°=140° and adjacent sides of 42 and 36 cm.

Using the cosine rule:
C²=A²+B²-2ABcos(α)
=42²+36²-2*42*36*cos(140°)
=1764+1296-3024(-.766044443)
=5376.52 approx.
C=sqrt(5376.52)
=73.32 cm (approx.)

To find the length of the longest diagonal of a parallelogram, we need to use the given information about the lengths of the adjacent sides and the angle.

First, let's label the parallelogram as ABCD, with AB = 42 cm and BC = 36 cm.

The opposite angles of a parallelogram are congruent, so angle ABC is also 40 degrees.

To find the length of diagonal AC, we can use the Law of Cosines, which states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle.

In this case, we consider triangle ABC, and we want to find AC^2.

By applying the Law of Cosines, we have:
AC^2 = AB^2 + BC^2 - 2(AB)(BC)cos(angle ABC)

Substituting the given values:
AC^2 = (42 cm)^2 + (36 cm)^2 - 2(42 cm)(36 cm)cos(40 degrees)

Calculating this equation will give us the square of the length of the longest diagonal.

AC^2 = 1764 cm^2 + 1296 cm^2 - 3024 cm^2*cos(40 degrees)

AC^2 = 3056 cm^2 - 3024 cm^2*(0.766)

AC^2 ≈ 3056 cm^2 - 2322.864 cm^2

AC^2 ≈ 733.136 cm^2

To find the length of AC, we take the square root of AC^2:
AC ≈ √733.136 cm
AC ≈ 27.1 cm (rounded to the nearest tenth of a centimeter)

Therefore, the measure of the longest diagonal of the parallelogram is approximately 27.1 cm.