Certainly! Let's solve the problem step-by-step to demonstrate that the distance between point [tex]\(O(2, 2)\)[/tex] and point [tex]\(A(5, 4)\)[/tex] is equal to the distance between point [tex]\(O(2, 2)\)[/tex] and point [tex]\(B(-1, 4)\)[/tex].
### Step-by-Step Solution:
#### (a) Calculate the distance [tex]\(OA\)[/tex] between points [tex]\(O\)[/tex] and [tex]\(A\)[/tex]:
Points:
[tex]\( O = (2, 2) \)[/tex]
[tex]\( A = (5, 4) \)[/tex]
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Using the coordinates of [tex]\(O\)[/tex] and [tex]\(A\)[/tex]:
[tex]\[ OA = \sqrt{(5 - 2)^2 + (4 - 2)^2} \][/tex]
Calculate the differences:
[tex]\[ x\text{-difference} = 5 - 2 = 3 \][/tex]
[tex]\[ y\text{-difference} = 4 - 2 = 2 \][/tex]
Square these differences:
[tex]\[ x\text{-difference squared} = 3^2 = 9 \][/tex]
[tex]\[ y\text{-difference squared} = 2^2 = 4 \][/tex]
Add these squares and take the square root:
[tex]\[ OA = \sqrt{9 + 4} = \sqrt{13} \approx 3.605551275463989 \][/tex]
#### (b) Calculate the distance [tex]\(OB\)[/tex] between points [tex]\(O\)[/tex] and [tex]\(B\)[/tex]:
Points:
[tex]\( O = (2, 2) \)[/tex]
[tex]\( B = (-1, 4) \)[/tex]
Using the coordinates of [tex]\(O\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[ OB = \sqrt{(-1 - 2)^2 + (4 - 2)^2} \][/tex]
Calculate the differences:
[tex]\[ x\text{-difference} = -1 - 2 = -3 \][/tex]
[tex]\[ y\text{-difference} = 4 - 2 = 2 \][/tex]
Square these differences:
[tex]\[ x\text{-difference squared} = (-3)^2 = 9 \][/tex]
[tex]\[ y\text{-difference squared} = 2^2 = 4 \][/tex]
Add these squares and take the square root:
[tex]\[ OB = \sqrt{9 + 4} = \sqrt{13} \approx 3.605551275463989 \][/tex]
#### (c) Compare the distances [tex]\(OA\)[/tex] and [tex]\(OB\)[/tex]:
We have:
[tex]\[ OA \approx 3.605551275463989 \][/tex]
[tex]\[ OB \approx 3.605551275463989 \][/tex]
Therefore, we can conclude:
[tex]\[ OA = OB \][/tex]
Hence, the distances from point [tex]\(O\)[/tex] to points [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are equal.
