# Find the measure of the longest diagonal to the nearest tenth of a centimeter:The lengths of two adjacent sides of a parallelogram are 42 cm and 36 cm. An angle of the parallelogram is 40°.

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It is given that the lengths of two adjacent sides of a parallelogram are 42 cm and 36 cm. One of the angles of the parallelogram is 40 degrees. The other angle is supplementary to 40 degrees, it is equal to 180 - 40 = 140 degrees.

Divide the parallelogram into 2 triangles. One has sides 36 and 42 and an enclosed angle of 40 degrees and the other has sides 36 and 42 and an enclosed angle of 140. The diagonal in both cases is the third side.

Use the cosine rule which states that c^2 = a^2 + b^2 - 2*a*b*cos C

c = sqrt [a^2 + b^2 - 2*a*b*cos C]

=> c = sqrt [ 36^2 + 42^2 - 2*36*42*cos 40]

=> 27.3 cm

and

c = sqrt [ 36^2 + 42^2 - 2*36*42*cos 140]

=> 73.3 cm

As 73.3 is larger than 27.3 that is the length of the longer diagonal.

**The length of the longer diagonal of the parallelogram is 73.3 cm**

The adjacent sides of the parallelogram are 42 and 36.

One othe angles of the parallelogram = 40 deg.

So the angle contained by the sides 42 and 36 is 40 or (180-40) = 140 deg.

Therefore the greatest diagonal is opposite to angle 140 with adjacent sides 42 and 36 or 36 and 42.

So the gretest diagonal = sqrt(36^2+42^2 - 2*36*42*cos140) using the cosine formula .

So greatest the diagonal = sqrt{1296+1764 - (-2316.52} = **73.32**.

This sum will require cos rule and trigonometrical tables.

let the //gm be ABCD. the longest diagonal will be BD if DC=42cm and BC=36cm

Let us see the cos rule:

In a triangle ABC,

Angle A is opp to side a(OR BC),

Angle B is opp to side b(OR AC)

Angle C is opp to side c(OR AB)

cos rule states that

c^2=a^2+b^2-2*a*b*cosC

now in Triangle BCD,

Angle BCD will be (180-40) 140 deg(and not 40 because we want longest diagonal,which should be opposite to greatest angle),therefore using cos rule

BD^2=BC^2+DC^2-2BC*DC*COS 140

BD^2=36^2+42^2-2*36*42*(-0.198) [from trigonometrical tables]

BD^2=1296+1764+598.75

BD^2=3658.75

BD=root of(3658.75)

BD=60.4876

BD=60.5cm(Ans)