Solved Distance Formula Midpoint Formula Pythagorean Theorem ... - Free Printable

Let's solve the problem step by step using the given information and formulas.

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Given:

- Ship A is at point $ A(7, 4) $
- Ship B is at point $ B(-9, -1) $
- They are to meet at the midpoint of the segment connecting them.
- Each grid unit = 400 miles

We are asked:

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a) Find the holodeck grid coordinates of the meeting location. Which formula is useful?

#### Step 1: Identify the correct formula
Since they are meeting at the midpoint of the line segment between the two ships, we use the Midpoint Formula:

$$
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

#### Step 2: Plug in the coordinates
Let:
- $ (x_1, y_1) = (7, 4) $ → Ship A
- $ (x_2, y_2) = (-9, -1) $ → Ship B

Now compute:

$$
x = \frac{7 + (-9)}{2} = \frac{-2}{2} = -1
$$
$$
y = \frac{4 + (-1)}{2} = \frac{3}{2} = 1.5
$$

So, the meeting point is at:
$$
\boxed{(-1, 1.5)}
$$

✔ Answer for part a):
- Coordinates of the meeting location: $ \boxed{(-1, 1.5)} $
- The useful formula is the Midpoint Formula.

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b) Find the holodeck distance between the ships’ current positions, to the nearest hundredth of a grid unit.

We need to find the distance between points $ A(7, 4) $ and $ B(-9, -1) $. Use the Distance Formula:

$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

Plug in values:

$$
d = \sqrt{(-9 - 7)^2 + (-1 - 4)^2} = \sqrt{(-16)^2 + (-5)^2}
$$
$$
= \sqrt{256 + 25} = \sqrt{281}
$$

Now calculate $ \sqrt{281} $:

$$
\sqrt{281} \approx 16.763
$$

Rounded to the nearest hundredth of a grid unit:
$$
\boxed{16.76} \text{ grid units}
$$

✔ Answer for part b):
- Distance between the ships: $ \boxed{16.76} $ grid units

> Note: This is in grid units. If you wanted actual miles, multiply by 400:
> $$
> 16.76 \times 400 = 6,704 \text{ miles}
> $$
> But the question asks for holodeck distance, so we keep it in grid units.

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✔ Final Answers:

a) Meeting location: $ \boxed{(-1, 1.5)} $
Useful formula: Midpoint Formula

b) Distance between ships: $ \boxed{16.76} $ grid units

Let me know if you'd like a visual explanation or further breakdown! 🚀