Find the angle in degree between adjacent sides of the pyramid PQRS. P(4,0,0), Q (4,4,0), R(0, 4,0), S (2, 2, 9). The base of the pyramid is not considered a side.

To find the angle between adjacent sides of the pyramid PQRS, we need to first find the vectors representing the sides of the pyramid.

Let's consider the sides PQ and PS.

Vector PQ = Q - P = (4, 4, 0) - (4, 0, 0) = (0, 4, 0)

Vector PS = S - P = (2, 2, 9) - (4, 0, 0) = (-2, 2, 9)

Next, we will normalize these vectors to find the direction vectors.

Magnitude of PQ = ||PQ|| = √(0^2 + 4^2 + 0^2) = √16 = 4

Normalized vector PQ = (0, 4, 0) / 4 = (0, 1, 0)

Magnitude of PS = ||PS|| = √((-2)^2 + 2^2 + 9^2) = √(4 + 4 + 81) = √89

Normalized vector PS = (-2, 2, 9) / √89

Now, we can find the dot product of these two direction vectors.

Dot product PQ · PS = (0, 1, 0) · (-2/√89, 2/√89, 9/√89)
= (0)(-2/√89) + (1)(2/√89) + (0)(9/√89)
= 2/√89

Next, we will find the magnitude of the dot product.

||PQ|| ||PS|| = 4 * √89

Now we can find the angle using the dot product formula:

cos θ = (PQ · PS) / (||PQ|| ||PS||)

θ = arccos((2/√89) / (4 * √89))

Using a calculator, we can approximate the angle to be around 59.25 degrees.

Therefore, the angle in degrees between the adjacent sides of the pyramid PQRS is approximately 59.25 degrees.

To find the angle in degrees between adjacent sides of the pyramid PQRS, we can use the dot product formula. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.

First, let's determine the two vectors corresponding to the adjacent sides of the pyramid PQRS. We can do this by subtracting the coordinates of the points:

Vector P to Q = (4, 4, 0) - (4, 0, 0) = (0, 4, 0)
Vector P to S = (2, 2, 9) - (4, 0, 0) = (-2, 2, 9)

Next, let's calculate the magnitudes of these vectors using the formula: magnitude = √(x^2 + y^2 + z^2).

Magnitude of P to Q = √(0^2 + 4^2 + 0^2) = √16 = 4
Magnitude of P to S = √((-2)^2 + 2^2 + 9^2) = √4 + 4 + 81 = √89

Now, let's calculate the dot product of these vectors using the formula: dot product = (x1 * x2) + (y1 * y2) + (z1 * z2).

Dot product of P to Q and P to S = (0 * -2) + (4 * 2) + (0 * 9) = 0 + 8 + 0 = 8

Using the dot product formula, we have:
dot product = |vector P to Q| * |vector P to S| * cos(angle)

Rearranging the formula, we can solve for the cosine of the angle:
cos(angle) = dot product / (|vector P to Q| * |vector P to S|)
cos(angle) = 8 / (4 * √89)
cos(angle) = 2 / √89

Finally, we can calculate the angle using the inverse cosine function (cos^(-1)) on a calculator:
angle = cos^(-1) (2 / √89)

Therefore, the angle in degrees between the adjacent sides of the pyramid PQRS can be found by evaluating this expression using a calculator.