Question

A parallelogram has sides of 18 and 26 ft, and an angle of 39° . Find the shorter diagonal

Answer 1

To find the shorter diagonal of a parallelogram with given side lengths and angle, use the formula √(a^2 + b^2 - 2abcosθ).

Step-by-step explanation:

To find the shorter diagonal of a parallelogram, we need to use the formula:

Shorter diagonal = √(a^2 + b^2 - 2abcosθ)

Where a and b are the sides of the parallelogram and θ is the angle between them.

Using the given information, we have a = 18 ft, b = 26 ft, and θ = 39°.

Substituting these values into the formula, we get:

Shorter diagonal = √(18^2 + 26^2 - 2(18)(26)cos(39°))

Calculating this expression gives us:

Shorter diagonal ≈ 9.15 ft

Answer 2

16.51

Step-by-step explanation:

In a parallelogram, the opposite angles are always equal in measure. So two of the angles in the parallelogram measure 39 degrees each.

The sum of angles of the parallelogram must be 360 degrees. Let the other two angles be x degree each. We can set up the following equation for the angles:

39 + 39 + x + x = 360

78 + 2x = 360

2x = 282

x = 141

This means, the other two angles measure 141 degree each. The shorter diagonal will be opposite to the shorter angle.

Hence, the diagonal opposite to the angle 39 degree will be the shorter one. A diagonal divides the parallelogram in two triangles. So we will have two sides and an included angle and we have to find the third side of the triangle which can be found using the law of cosines. Let the third side be c as shown in image below, using the law of cosines, we can write:

`c^(2) = a^(2)+ b^(2) -2ab cos(\gamma)\\\\c^(2)=18^(2)+26^(2)-2(18)(26)cos(39)\\\\ c^(2)=272.59\\\\ c=16.51`

Thus the shorter diagonal will be 16.51 feet in measure.