Answer :
Sure! Let's go through the detailed step-by-step solution for finding the distances for the points given in the table. Each distance is calculated from the origin (0,0) to the given point using the distance formula:
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Since we are calculating the distance from the origin [tex]\((0, 0)\)[/tex] to a point [tex]\((a, b)\)[/tex], the formula simplifies to:
[tex]\[ d = \sqrt{a^2 + b^2} \][/tex]
Let's find the distances corresponding to [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex] for each of the remaining points in the table.
### 1. Calculating Distance [tex]\(X\)[/tex]
For the point [tex]\((-1, -2)\)[/tex]:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = -2 \][/tex]
[tex]\[ X = \sqrt{(-1)^2 + (-2)^2} \][/tex]
[tex]\[ X = \sqrt{1 + 4} \][/tex]
[tex]\[ X = \sqrt{5} \][/tex]
[tex]\[ X \approx 2.23606797749979 \][/tex]
### 2. Calculating Distance [tex]\(Y\)[/tex]
For the point [tex]\((-4, 1)\)[/tex]:
[tex]\[ a = -4 \][/tex]
[tex]\[ b = 1 \][/tex]
[tex]\[ Y = \sqrt{(-4)^2 + 1^2} \][/tex]
[tex]\[ Y = \sqrt{16 + 1} \][/tex]
[tex]\[ Y = \sqrt{17} \][/tex]
[tex]\[ Y \approx 4.123105625617661 \][/tex]
### 3. Calculating Distance [tex]\(Z\)[/tex]
For the point [tex]\((-6, -3)\)[/tex]:
[tex]\[ a = -6 \][/tex]
[tex]\[ b = -3 \][/tex]
[tex]\[ Z = \sqrt{(-6)^2 + (-3)^2} \][/tex]
[tex]\[ Z = \sqrt{36 + 9} \][/tex]
[tex]\[ Z = \sqrt{45} \][/tex]
[tex]\[ Z \approx 6.708203932499369 \][/tex]
In conclusion, using the distance formula for the given points, we find that:
- Distance [tex]\(X\)[/tex] for the point [tex]\((-1, -2)\)[/tex] is approximately [tex]\(2.23606797749979\)[/tex].
- Distance [tex]\(Y\)[/tex] for the point [tex]\((-4, 1)\)[/tex] is approximately [tex]\(4.123105625617661\)[/tex].
- Distance [tex]\(Z\)[/tex] for the point [tex]\((-6, -3)\)[/tex] is approximately [tex]\(6.708203932499369\)[/tex].
These are the distances from the origin to the given points in the table.
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Since we are calculating the distance from the origin [tex]\((0, 0)\)[/tex] to a point [tex]\((a, b)\)[/tex], the formula simplifies to:
[tex]\[ d = \sqrt{a^2 + b^2} \][/tex]
Let's find the distances corresponding to [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex] for each of the remaining points in the table.
### 1. Calculating Distance [tex]\(X\)[/tex]
For the point [tex]\((-1, -2)\)[/tex]:
[tex]\[ a = -1 \][/tex]
[tex]\[ b = -2 \][/tex]
[tex]\[ X = \sqrt{(-1)^2 + (-2)^2} \][/tex]
[tex]\[ X = \sqrt{1 + 4} \][/tex]
[tex]\[ X = \sqrt{5} \][/tex]
[tex]\[ X \approx 2.23606797749979 \][/tex]
### 2. Calculating Distance [tex]\(Y\)[/tex]
For the point [tex]\((-4, 1)\)[/tex]:
[tex]\[ a = -4 \][/tex]
[tex]\[ b = 1 \][/tex]
[tex]\[ Y = \sqrt{(-4)^2 + 1^2} \][/tex]
[tex]\[ Y = \sqrt{16 + 1} \][/tex]
[tex]\[ Y = \sqrt{17} \][/tex]
[tex]\[ Y \approx 4.123105625617661 \][/tex]
### 3. Calculating Distance [tex]\(Z\)[/tex]
For the point [tex]\((-6, -3)\)[/tex]:
[tex]\[ a = -6 \][/tex]
[tex]\[ b = -3 \][/tex]
[tex]\[ Z = \sqrt{(-6)^2 + (-3)^2} \][/tex]
[tex]\[ Z = \sqrt{36 + 9} \][/tex]
[tex]\[ Z = \sqrt{45} \][/tex]
[tex]\[ Z \approx 6.708203932499369 \][/tex]
In conclusion, using the distance formula for the given points, we find that:
- Distance [tex]\(X\)[/tex] for the point [tex]\((-1, -2)\)[/tex] is approximately [tex]\(2.23606797749979\)[/tex].
- Distance [tex]\(Y\)[/tex] for the point [tex]\((-4, 1)\)[/tex] is approximately [tex]\(4.123105625617661\)[/tex].
- Distance [tex]\(Z\)[/tex] for the point [tex]\((-6, -3)\)[/tex] is approximately [tex]\(6.708203932499369\)[/tex].
These are the distances from the origin to the given points in the table.
