Question

use the distance formula to find the distance between each pair of points.

1. ( u(0, 1) ) and ( v(-3, -9) )
2. ( m(10, -1) ) and ( n(2, -5) )
3. ( p(-10, 1) ) and ( q(5, 5) )
4. ( f(6, 15) ) and ( g(4, 24) )
5. astronomy each unit on the map of a section of a moon represents 1 kilometer. to the nearest tenth of a kilometer, what is the distance between the two craters?

(there is a coordinate grid with crater a at (25, 85) and crater b at (60, 15))

Explanation:

Problem 16: \( U(0, 1) \) and \( V(-3, -9) \)

Step 1: Recall the Distance Formula

The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step 2: Identify the coordinates

Here, \( x_1 = 0 \), \( y_1 = 1 \), \( x_2 = -3 \), \( y_2 = -9 \).

Step 3: Substitute into the formula

\[

$$\begin{align*} d &= \sqrt{(-3 - 0)^2 + (-9 - 1)^2}\\ &= \sqrt{(-3)^2 + (-10)^2}\\ &= \sqrt{9 + 100}\\ &= \sqrt{109}\\ &\approx 10.4 \end{align*}$$

\]

Step 1: Use the Distance Formula

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

Step 2: Identify coordinates

\( x_1 = 10 \), \( y_1 = -1 \), \( x_2 = 2 \), \( y_2 = -5 \)

Step 3: Substitute values

\[

$$\begin{align*} d &= \sqrt{(2 - 10)^2 + (-5 - (-1))^2}\\ &= \sqrt{(-8)^2 + (-4)^2}\\ &= \sqrt{64 + 16}\\ &= \sqrt{80}\\ &\approx 8.9 \end{align*}$$

\]

Step 1: Apply Distance Formula

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

Step 2: Identify coordinates

\( x_1 = -10 \), \( y_1 = 1 \), \( x_2 = 5 \), \( y_2 = 5 \)

Step 3: Substitute and calculate

\[

$$\begin{align*} d &= \sqrt{(5 - (-10))^2 + (5 - 1)^2}\\ &= \sqrt{(15)^2 + (4)^2}\\ &= \sqrt{225 + 16}\\ &= \sqrt{241}\\ &\approx 15.5 \end{align*}$$

\]

Answer:

\( \approx 10.4 \)

Problem 17: \( M(10, -1) \) and \( N(2, -5) \)