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

Question

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

Wyatt Anderson · Accepted Answer

Longest Diagonal: #color(magenta)(22.99)# (approx.)

Extending the side with length #15# to form a right triangle as in the image below:
#color(white)("XXXX")=sqrt(8^2-(8/15)^2) ~~7.892#

And the longest diagonal of the original parallelogram is
#color(white)("XXX")sqrt((15+color(green)(x))^2+color(blue)(h)^2)#

#color(white)("XXX")~~22.99#

Sadie Allen · Answer

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

Longest diagonal = √(a^2 + b^2 + 2abcos(θ))

Where:
- $a$ and $b$ are the lengths of the sides of the parallelogram.
- $θ$ is the angle between the two sides with lengths $a$ and $b$.

Given that the sides of the parallelogram are 15 and 8, and its area is 8, we can use the area formula of a parallelogram to find the angle between the sides:

Area = base × height
8 = 15 × height
height = 8/15

Now, using the Law of Cosines, we find the angle $θ$ between the sides:

$θ = \cos^{-1}\left(\frac{8^2 + 15^2 - d^2}{2 \times 8 \times 15}\right)$

Where $d$ is the length of the longest diagonal.

Given that the area is 8, the product of the height and the length of the base is 8. So, the height of the parallelogram is 8/15.

Using the Law of Cosines, we can find the angle $θ$ between the sides. Then, using the formula for the longest diagonal, we can calculate its length.

Substitute the values into the formula and solve for the longest diagonal, $d$.

Wyatt Anderson · Answer

undefined To find the length of the longest diagonal of the parallelogram, we can use the formula for the area of a parallelogram:

$$ \text{Area} = \text{base} \times \text{height} $$

Given that the area is 8 and one of the sides (base) is 15, we can solve for the height:

$$ 8 = 15 \times \text{height} $$

$$ \text{height} = \frac{8}{15} $$

Now, using the height and the other side (8) as the base, we can find the length of the longest diagonal using the Pythagorean theorem:

$$ \text{Longest diagonal}^2 = \text{base}^2 + \text{height}^2 $$

$$ \text{Longest diagonal}^2 = 8^2 + \left(\frac{8}{15}\right)^2 $$

$$ \text{Longest diagonal}^2 = 64 + \frac{64}{225} $$

$$ \text{Longest diagonal}^2 = \frac{14400 + 64}{225} $$

$$ \text{Longest diagonal}^2 = \frac{14464}{225} $$

$$ \text{Longest diagonal} = \sqrt{\frac{14464}{225}} $$

$$ \text{Longest diagonal} \approx \sqrt{64.29} $$

$$ \text{Longest diagonal} \approx 8.01 $$

So, the length of the longest diagonal of the parallelogram is approximately 8.01 units.