Let's work through some of the problems in this image.

Problem 15

We are asked to find the distance between two points $K(0, 5)$ and $L(7, 9)$. The distance formula is:

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

Substitute the values:

$d = \sqrt{(7 - 0)^2 + (9 - 5)^2} = \sqrt{7^2 + 4^2} = \sqrt{49 + 16} = \sqrt{65}$

The square root of 65 is approximately 8.1, so the distance is 8.1 units.

Problem 16

For points $C(-2, 6)$ and $D(10, -8)$, using the same formula:

$d = \sqrt{(10 - (-2))^2 + (-8 - 6)^2} = \sqrt{(10 + 2)^2 + (-8 - 6)^2} = \sqrt{12^2 + (-14)^2} = \sqrt{144 + 196} = \sqrt{340}$

The square root of 340 is approximately 18.4, so the distance is 18.4 units.

Problem 17

For points $W(-3, -5)$ and $X(-9, 4)$:

$d = \sqrt{(-9 - (-3))^2 + (4 - (-5))^2} = \sqrt{(-9 + 3)^2 + (4 + 5)^2} = \sqrt{(-6)^2 + 9^2} = \sqrt{36 + 81} = \sqrt{117}$

The square root of 117 is approximately 10.8, so the distance is 10.8 units.

Problem 18

For points $G(-12, -11)$ and $H(5, -9)$:

$d = \sqrt{(5 - (-12))^2 + (-9 - (-11))^2} = \sqrt{(5 + 12)^2 + (-9 + 11)^2} = \sqrt{17^2 + 2^2} = \sqrt{289 + 4} = \sqrt{293}$

The square root of 293 is approximately 17.1, so the distance is 17.1 units.

Problem 19

For points $A(6, 2)$ and $B(-3, -1)$:

$d = \sqrt{(-3 - 6)^2 + (-1 - 2)^2} = \sqrt{(-9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90}$

The square root of 90 is approximately 9.5, so the distance is 9.5 units.

Would you like more details or have any specific questions? Here are some additional questions related to this topic:

1. What is the midpoint formula, and how can it be applied to find the midpoint between two points?
2. Can the distance formula be derived from the Pythagorean theorem? How?
3. How do you calculate distances in three-dimensional space?
4. What are some applications of the distance formula in real-life problems?
5. How can you simplify square roots like the ones we found here?

Tip:

Always double-check your signs (positive or negative) when using the distance formula, as small errors can lead to incorrect results.