# A parallelogram has sides with lengths of #5 # and #9 #. If the parallelogram's area is #45 #, what is the length of its longest diagonal?

Diagonal length:

By signing up, you agree to our Terms of Service and Privacy Policy

I am advised that the found condition is mathematically compatible with the shape being a rectangle.

When in doubt do a very quick and rough sketch in the margin. For this question it should take about 5 to 6 seconds

Notice that

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Determine the length of DE (h) using area

Determine the length of FE and hence AE

Solve for AD using Pythagoras

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If you were to get a pair of scissors and cut out triangle ABG you will discover that it will fit exactly over the empty space DEF. The result being a rectangle.

The area of the rectangle is:

Divide both sides by

But

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now we find out that

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Greatest slope length is the diagonal of the rectangle.

Let the slope length be

Both 2 and 56 are prime numbers so we can not simplify any further

By signing up, you agree to our Terms of Service and Privacy Policy

To find the length of the longest diagonal of the parallelogram, you can use the formula:

( \text{Area} = \text{Base} \times \text{Height} )

Given that the area is 45 and one side is 5, you can solve for the height:

( 45 = 5 \times \text{Height} )

( \text{Height} = \frac{45}{5} )

( \text{Height} = 9 )

Now, you can use the Pythagorean theorem to find the length of the longest diagonal:

( \text{Diagonal}^2 = 5^2 + 9^2 )

( \text{Diagonal}^2 = 25 + 81 )

( \text{Diagonal}^2 = 106 )

( \text{Diagonal} = \sqrt{106} )

So, the length of the longest diagonal is approximately ( \sqrt{106} ) units.

By signing up, you agree to our Terms of Service and Privacy Policy

- Two rhombuses have sides with lengths of #2 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #pi/4 #, what is the difference between the areas of the rhombuses?
- Two opposite sides of a parallelogram have lengths of #8 #. If one corner of the parallelogram has an angle of #pi/8 # and the parallelogram's area is #24 #, how long are the other two sides?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #6 # and sides C and D have a length of # 4 #. If the angle between sides A and C is #(7 pi)/12 #, what is the area of the parallelogram?
- Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(3pi)/4 #, what is the difference between the areas of the rhombuses?
- A parallelogram has sides with lengths of #7 # and #15 #. If the parallelogram's area is #21 #, what is the length of its longest diagonal?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7