730 ft.

To determine the distance the shortstop needs to throw the ball to first base, we can use the coordinates of the bases on a standard baseball diamond, where each base is 90 feet from each other, and the diamond forms a square.

Let's define the positions of the bases:

• Second base is at (90, 90).
• First base is at (0, 90).
• The shortstop is standing on the base path between second and third base, specifically 30 feet away from second base toward third base.

Given that the distance from second base (90, 90) to the shortstop is 30 feet directly along the base path toward third base:

• The shortstop's position will be at (90, 60) since she is moving down the y-coordinate by 30 feet.

Now, we need to find the distance from the shortstop (90, 60) to first base (0, 90). We can use the distance formula:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting in our coordinates:

• \(x_1 = 90\), \(y_1 = 60\) (shortstop's position)
• \(x_2 = 0\), \(y_2 = 90\) (first base)

\[ \text{Distance} = \sqrt{(0 - 90)^2 + (90 - 60)^2} \] \[ = \sqrt{(-90)^2 + (30)^2} \] \[ = \sqrt{8100 + 900} \] \[ = \sqrt{9000} \] \[ = 30\sqrt{10} \]

Now, calculating \(30\sqrt{10}\):

\[ \sqrt{10} \approx 3.162 \] \[ 30\sqrt{10} \approx 30 \times 3.162 = 94.86 \]

Rounding to the nearest tenth gives:

\[ \text{Distance} \approx 94.9 \text{ feet} \]

So, the shortstop needs to throw the ball approximately 94.9 feet to first base.

Apply the Pythagorean Theorem to find the length between the two points. (7,8) (3,2)
Round your answer to the nearest hundredth.
(1 point)

To find the distance between the two points \((7, 8)\) and \((3, 2)\) using the Pythagorean Theorem, we can first determine the difference in the x-coordinates and the y-coordinates:

\[ \Delta x = x_2 - x_1 = 3 - 7 = -4 \] \[ \Delta y = y_2 - y_1 = 2 - 8 = -6 \]

Now we can apply the Pythagorean theorem, which states that the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:

\[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]

Substituting the values we found:

\[ d = \sqrt{(-4)^2 + (-6)^2} \] \[ = \sqrt{16 + 36} \] \[ = \sqrt{52} \]

Now, to simplify \(\sqrt{52}\):

\[ \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \]

Calculating \(2\sqrt{13}\):

\[ \sqrt{13} \approx 3.60555 \] \[ 2\sqrt{13} \approx 2 \times 3.60555 \approx 7.2111 \]

Rounding to the nearest hundredth gives:

\[ d \approx 7.21 \]

Thus, the distance between the points \((7, 8)\) and \((3, 2)\) is approximately 7.21 units.

15
16
Find the slant height of the square pyramid.
(1 point)

To find the slant height of a square pyramid, we need to know the height of the pyramid and the length of the base. The slant height is the length from the apex (top point) of the pyramid to the midpoint of one of the sides of the base.

Given:

• The base of the pyramid is square.
• Let's assume the side length of the base is \(b\) and the height of the pyramid (the vertical distance from the base to the apex) is \(h\).

The formula for the slant height (\(l\)) of a square pyramid can be derived using the Pythagorean theorem:

1. Calculate the apex to the midpoint of a side of the base. Since the base is square, the distance from the center of the base to the midpoint of one side is \(\frac{b}{2}\).

2. To find the slant height, use the Pythagorean theorem:

\[ l = \sqrt{h^2 + \left(\frac{b}{2}\right)^2} \]

If we assume:

• The height \(h = 16\)
• The base side length \(b = 15\)

Plugging in these values:

\[ l = \sqrt{16^2 + \left(\frac{15}{2}\right)^2} \] \[ = \sqrt{256 + \left(7.5\right)^2} \] \[ = \sqrt{256 + 56.25} \] \[ = \sqrt{312.25} \] \[ \approx 17.66 \]

Thus, the slant height of the square pyramid is approximately 17.66 units.

S: 1,2,3,4
T: 6,5,7,22
In the picture, one angle is labeled 22° and lines S and T' are parallel.
Which other three angles will be 22°? Separate each angle number with a comma.
(1 point)

Unfortunately, without a specific diagram to reference, I can provide a general explanation based on the concepts of parallel lines and angles formed by a transversal.

If lines S and T' are parallel and one angle is labeled 22°, the other angles that would also measure 22° are typically formed depending on their relationship with the transversal and the given angle.

Here are the possible relationships:

1. Corresponding Angles: If one angle is 22°, its corresponding angle on the same side of the transversal will also be 22°.

2. Alternate Interior Angles: If one angle is 22°, the alternate interior angle (the angle that lies on the opposite side of the transversal but within the two parallel lines) will also be 22°.

3. Consecutive Interior Angles: If one angle is 22°, the interior angles on the same side of the transversal will not necessarily be 22°, while they are supplementary to any angle opposite to them.

Given these relationships, if the labeled angle is 22°, the corresponding angles formed will also be 22°. You would typically label these angles based on their positions (e.g., angle 1, angle 2, angle 3).

If you have specific angle labels assigned to the diagram, please refer to that and determine through angles that are either corresponding or alternate interior to the 22° angle. If you could clarify or provide angle numbers, I could help identify those specifically!