To determine whether the points [tex]\((9, 5)\)[/tex] and [tex]\((5, 9)\)[/tex] are at the same distance from the origin [tex]\((0, 0)\)[/tex], we use the Distance Formula. The Distance Formula, which calculates the distance [tex]\(d\)[/tex] 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 each point to the origin, the formula can be simplified to:
[tex]\[ d = \sqrt{x^2 + y^2} \][/tex]
### Part (a)
1. Calculate the distance of the point [tex]\((9, 5)\)[/tex] from the origin:
Using the coordinates [tex]\((x, y) = (9, 5)\)[/tex]:
[tex]\[
d = \sqrt{9^2 + 5^2}
\][/tex]
[tex]\[
d = \sqrt{81 + 25}
\][/tex]
[tex]\[
d = \sqrt{106}
\][/tex]
The approximate value of [tex]\(\sqrt{106}\)[/tex] is:
[tex]\[
d \approx 10.295630140987
\][/tex]
Therefore, the distance from the point [tex]\((9, 5)\)[/tex] to the origin is approximately:
[tex]\[
10.295630140987
\][/tex]
2. Calculate the distance of the point [tex]\((5, 9)\)[/tex] from the origin:
Using the coordinates [tex]\((x, y) = (5, 9)\)[/tex]:
[tex]\[
d = \sqrt{5^2 + 9^2}
\][/tex]
[tex]\[
d = \sqrt{25 + 81}
\][/tex]
[tex]\[
d = \sqrt{106}
\][/tex]
The approximate value of [tex]\(\sqrt{106}\)[/tex] is:
[tex]\[
d \approx 10.295630140987
\][/tex]
Therefore, the distance from the point [tex]\((5, 9)\)[/tex] to the origin is approximately:
[tex]\[
10.295630140987
\][/tex]
### Conclusion
Both points [tex]\((9, 5)\)[/tex] and [tex]\((5, 9)\)[/tex] are at the same distance from the origin, and the distance is approximately [tex]\(10.295630140987\)[/tex].
