(5,0) and (-9,9) find the distance between the two points in simplest radical form

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:

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

Given points are \((5, 0)\) and \((-9, 9)\). Let's plug these values into the formula:

\[
d = \sqrt{(-9 - 5)^2 + (9 - 0)^2}
\]

Now, calculate each part:

1. Calculate \((-9 - 5)\):

   \[
   -9 - 5 = -14
   \]

2. Calculate \((9 - 0)\):

   \[
   9 - 0 = 9
   \]

Substituting back into the formula:

\[
d = \sqrt{(-14)^2 + (9)^2}
\]

\[
d = \sqrt{196 + 81}
\]

\[
d = \sqrt{277}
\]

So, the distance between the points \((5, 0)\) and \((-9, 9)\) in simplest radical form is:

\[
\boxed{\sqrt{277}}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you simplify a square root?
2. What is the midpoint of the line segment between the two points?
3. How can you determine if two line segments are of equal length?
4. How do you find the slope of the line connecting two points?
5. What is the equation of the line passing through these two points?

**Tip:** Always simplify inside the radical first before calculating to keep the steps manageable.

Find the distance between the points (5,0) and (-9,9) in simplest radical form.

Calculate the distance between the points (5,0) and (-9,9) using the distance formula. This problem involves key concepts in coordinate geometry, including applying the Pythagorean theorem. Ideal for students in Grades 8-10.

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