We are asked to explain how the fact that two points are on opposite sides of the $y -$ axis affects the process of finding the distance between the two points. Let's consider an example pair of points like these.

To find the distance between the two points, we can draw a right triangle where the segment connecting our points is the hypotenuse.

We need to find the lengths of the legs of our triangle. To find the length of the side $A C,$ we subtract the absolute values of the $y -$ coordinate of point $C$ from the absolute value of the $y -$ coordinate of point $A .$

Next, we find the length of the side $B C .$ This time, we use the absolute values of the $x -$ coordinates of the points. However, since the points are on opposite sides of the $y -$ axis, we add the absolute values instead of subtracting them.

Now that we know the lengths of the legs of our triangle, we can use the Pythagorean Theorem to find the distance between $A$ and $B .$ Let's call this distance $c .$

The Pythagorean Theorem tells us that the square of length

c,

the hypotenuse of our right triangle, is equal to the sum of the squares of the lengths of the legs of the triangle. This lets us write an equation.

c^{2} = 4^{2} + 3^{2}

Let's solve this equation!

c^{2} = 4^{2} + 3^{2}

CalcPow

Calculate power

c^{2} = 16 + 9

AddTerms

Add terms

c^{2} = 25

SqrtEqn

$LHS = RHS$

c^{2} = 25

RootPowToNumber

$a = a$

c = 25

CalcRoot

Calculate root

c = 5

The distance between the points is

5 .

The fact that the points are on opposite sides of the

y -

axis affected the process by making us add the absolute values of the coordinates while finding the length of the horizontal leg of the right triangle instead of subtracting them.

Exercise

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