# Two opposite sides of a parallelogram have lengths of #5 #. If one corner of the parallelogram has an angle of #pi/4 # and the parallelogram's area is #45 #, how long are the other two sides?

Here is a diagram.

The area of a parallelogram is given by

We know the base and the area, as shows the diagram above. However, we need to find the height.

Now, we can use right angled trig to find the length of the other side. We know the side opposite

By special angles,

Therefore,

So, the other two sides measure

Hopefully this helps!

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Given that two opposite sides of a parallelogram have lengths of 5, one corner of the parallelogram has an angle of π/4, and the parallelogram's area is 45, we can find the lengths of the other two sides using the formula for the area of a parallelogram:

Area = base * height

The base of the parallelogram is one of the sides whose length is 5, and the height is the distance between this side and the opposite side.

Let's denote the length of the other side as b and the height as h.

We know that the area is 45, and the base is 5, so we can rewrite the formula as:

45 = 5 * h

Solving for h:

h = 45 / 5 = 9

Now, we have the height of the parallelogram. To find the lengths of the other two sides, we need to use trigonometry.

Since the angle at one corner of the parallelogram is π/4, and opposite angles in a parallelogram are equal, the other angle at the same corner is also π/4.

Using trigonometry, we can find the lengths of the other two sides:

[ \text{Adjacent side} = \text{Hypotenuse} \times \cos(\text{angle}) ]

[ \text{Adjacent side} = 5 \times \cos(\pi/4) ]

[ \text{Adjacent side} = 5 \times \frac{\sqrt{2}}{2} ]

[ \text{Adjacent side} = \frac{5\sqrt{2}}{2} ]

So, the lengths of the other two sides are ( \frac{5\sqrt{2}}{2} ).

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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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