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Triangles Unit Test

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An illustration shows a square with smaller squares representing baseball plates on the top left, top right, and bottom right vertices. The bottom left vertex has a pentagon that represents home plate. A stick figure is located on the top of the square and the measurement from the right vertex to the stick figure is labeled 30 feet. A dashed diagonal line runs from the stick figure to the inner right side of the square, forming the hypotenuse of a triangle. The top right side of the square forms the right angle of the triangle and is marked with a right angle symbol. The outer right side of the square is labeled 90 feet.

A shortstop is standing in the base path between second and third base when she fields the ball. She is 30 feet from second base and needs to make the throw to first base. Assuming the field is square, how far does she need to throw the ball? Round your answer to the nearest tenth.

(1 point)
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1 answer

To solve the problem, let's analyze the situation using the information given.

  1. Vertices:

    • Assume the square baseball field has its bases positioned at the vertices. Let:
      • A (1st Base) be at (0, 0)
      • B (2nd Base) be at (90, 0)
      • C (3rd Base) be at (90, 90)
      • D (Home Plate) be at (0, 90)

  2. Position of the Shortstop:

    • The shortstop is in the base path between second base (B) and third base (C) and is 30 feet from second base.
    • Therefore, her position can be calculated as:
      • 30 feet towards third base means she is positioned at (90, 0) + (0, 30) = (90, 30).

  3. First Base Position:

    • First base is at (0, 0) as assumed above.

  4. Calculating the distance:

    • To find the distance from the shortstop’s position at (90, 30) to first base (0, 0), we use the distance formula:

    \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (90, 30)\).

    Plugging in the values, we get: \[ \text{Distance} = \sqrt{(90 - 0)^2 + (30 - 0)^2} \] \[ = \sqrt{90^2 + 30^2} \] \[ = \sqrt{8100 + 900} \] \[ = \sqrt{9000} \] \[ = 30\sqrt{10} \]

    Approximating \(\sqrt{10} \approx 3.162\): \[ 30 \times 3.162 \approx 94.86 \]

  5. Rounding:

    • Rounding to the nearest tenth, the distance the shortstop needs to throw the ball is approximately 94.9 feet.

Therefore, the final answer is: 94.9 feet.