Question

1. you are given the following points on a coordinate plane: ( x(-3,3.3) ), ( y(-3,-5.2) ), and ( z(4.5,-5.2) ).

part a
using absolute value, find the distance (number of units) between points ( x ) and ( y ).
part b
select all the coordinates that are 5 units from point ( z ).
( (-0.5,-10.2) )
( (4.5,-0.2) )
( (-0.5,-5.2) )
( (9.5,-0.2) )
( (4.5,-5.2) )

Explanation:

Step1: Calculate the distance between \(X(-3,3.3)\) and \(Y(-3, - 5.2)\)

Since the \(x -\)coordinates of \(X\) and \(Y\) are the same (\(x=-3\)), we use the formula for the distance between two points with the same \(x -\)coordinate \(d=\vert y_1 - y_2\vert\).
Here, \(y_1 = 3.3\) and \(y_2=-5.2\), so \(d=\vert3.3-(-5.2)\vert=\vert3.3 + 5.2\vert=\vert8.5\vert = 8.5\)

Step2: Check the distance from \(Z(4.5,-5.2)\) to each point in Part B

• For the point \((-0.5,-10.2)\):

Since the \(x -\)coordinates are different and \(y -\)coordinates are different. But if we consider the vertical distance (since \(x = 4.5\) and \(x=-0.5\), \(\vert4.5-(-0.5)\vert=5\), and \(\vert- 5.2-(-10.2)\vert=\vert-5.2 + 10.2\vert = 5\). Using the Pythagorean theorem \(d=\sqrt{(4.5+0.5)^2+(-5.2 + 10.2)^2}=\sqrt{5^2+5^2}=\sqrt{25 + 25}=\sqrt{50}
eq5\)

• For the point \((4.5,-0.2)\):

Since the \(x -\)coordinates are the same (\(x = 4.5\)), we use \(d=\vert y_1 - y_2\vert\). Here, \(y_1=-5.2\) and \(y_2=-0.2\), so \(d=\vert-5.2-(-0.2)\vert=\vert-5.2 + 0.2\vert=\vert-5\vert = 5\)

• For the point \((-0.5,-5.2)\):

Since the \(y -\)coordinates are the same (\(y=-5.2\)), we use \(d=\vert x_1 - x_2\vert\). Here, \(x_1 = 4.5\) and \(x_2=-0.5\), so \(d=\vert4.5-(-0.5)\vert=\vert4.5 + 0.5\vert=5\)

• For the point \((9.5,-0.2)\):

\(d=\sqrt{(4.5 - 9.5)^2+(-5.2+0.2)^2}=\sqrt{(-5)^2+(-5)^2}=\sqrt{25 + 25}=\sqrt{50}
eq5\)

• For the point \((4.5,-5.2)\):

\(d=\vert-5.2-(-5.2)\vert+\vert4.5 - 4.5\vert=0
eq5\)

Answer:

Part A: The distance between \(X\) and \(Y\) is \(8.5\) units.
Part B: The coordinates that are \(5\) units from \(Z\) are \((4.5,-0.2)\) and \((-0.5,-5.2)\)