Math  /  Geometry

QuestionQuestion Find the distance between the two points in simplest radical form. (8,3) and (1,4)(-8,-3) \text { and }(-1,4)
Answer Attempt 1 out of 2

Studdy Solution

STEP 1

What is this asking? We need to find the length of the straight line connecting two points on a graph! Watch out! Don't forget to take the square root at the end to get the actual distance, and remember to simplify that radical!

STEP 2

1. Visualize and strategize
2. Calculate the changes
3. Apply the distance formula

STEP 3

Alright, imagine these two points hanging out on a graph.
We've got (8,3)(-8, -3) and (1,4)(-1, 4).
We want to find the distance between them, which is like finding the length of a secret tunnel connecting them!

STEP 4

We can draw an imaginary right triangle using these points.
The distance between the points is the hypotenuse of that triangle.
The Pythagorean theorem is our tool of choice here!

STEP 5

Let's find the lengths of the legs of our imaginary triangle.
These lengths are just the **changes** in the xx and yy coordinates between our two points.

STEP 6

The change in xx is the difference between the xx coordinates: (1)(8)=1+8=7(-1) - (-8) = -1 + 8 = \mathbf{7}.

STEP 7

The change in yy is the difference between the yy coordinates: 4(3)=4+3=74 - (-3) = 4 + 3 = \mathbf{7}.

STEP 8

The distance formula is just a fancy way of writing the Pythagorean theorem!
It says: distance=(change in x)2+(change in y)2distance = \sqrt{(\text{change in } x)^2 + (\text{change in } y)^2}.

STEP 9

Let's plug in our **changes**! distance=(7)2+(7)2distance = \sqrt{(\mathbf{7})^2 + (\mathbf{7})^2}.

STEP 10

Time to calculate! distance=49+49=98distance = \sqrt{49 + 49} = \sqrt{98}.

STEP 11

Now, we need to simplify this radical.
We're looking for perfect squares that are factors of 98\mathbf{98}.
We know that 98=49298 = 49 \cdot 2, and 4949 is a perfect square!

STEP 12

So, 98=492=492=72\sqrt{98} = \sqrt{49 \cdot 2} = \sqrt{49} \cdot \sqrt{2} = 7\sqrt{2}.
Awesome!

STEP 13

The distance between the points (8,3)(-8, -3) and (1,4)(-1, 4) is 727\sqrt{2}.

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