A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #12 #, what is the length of its longest diagonal?

Answer 1

Length of longest diagonal #22.974# (approx.)

Using a side of length #8# as the base,
since the area is #12#
the height relative to a side of length #8# is #color(purple)(h=12/8=3/2)#

The required diagonal (#color(red)(d)#) is
the hypotenuse of a right triangle formed by extending the base by an amount #color(purple)(x)# until the line segment terminates at a point which is at a right angle under the furthest upper vertex of the parallelogram.

The length of this extension (#color(purple)(x)#) can be calculated using the Pythagorean Theorem:
#color(white)("XXX")color(purple)x=sqrt(15^2-h^2)~~14.928# (using a calculator)

The requested diagonal now can also be calculated using the Pythagorean Theorem as:
#color(white)("XXX")color(red)d=sqrt((8+x)^2+h^2) ~~22.974# (again with a calculator)

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Answer 2

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

( \text{Longest diagonal} = \sqrt{(a^2 + b^2 + 2ab \cdot \cos(\theta))} )

Where:

  • ( a ) and ( b ) are the lengths of the sides of the parallelogram,
  • ( \theta ) is the angle between the sides.

Given that the sides of the parallelogram have lengths of 15 and 8, and the area is 12, you can calculate the angle between the sides using the formula:

( \text{Area} = ab \cdot \sin(\theta) )

By substituting the given values, you can solve for ( \theta ). Once you have ( \theta ), you can find the length of the longest diagonal using the formula for the longest diagonal.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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