(b) The area of a parallelogram is 48 cm². If the two adjacent sides are 8 cm and 6 cm, find the length of its diagonal.​

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  10 cm

Step-by-step explanation:

The given area is the product of the side lengths, so the angle between them must be 90°.

  Area = ab·sin(C) . . . . . where C is the angle between sides 'a' and 'b'.

  48 cm² = (8 cm)(6 cm)·sin(C)

  1 = sin(C)   ⇒   C = 90° . . . . . . . . divide by 48 cm², find the arcsin

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The diagonal of the parallelogram (rectangle) can be found from the Pythagorean theorem. (Using the law of cosines would give the same result.)

  d² = 8² +6² = 64+36 = 100

  d = √100 = 10 . . . . cm

The length of the diagonal is 10 cm.

Additional comment

You may recognize the diagonal and the given sides form a 3-4-5 right triangle with a scale factor of 2. The diagonal will be the hypotenuse of a right triangle only if the parallelogram is a rectangle.

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