To find the value of [tex]\( p \)[/tex] given that the distance between the points [tex]\((1,2)\)[/tex] and [tex]\((16,p)\)[/tex] is 17, we can use the distance formula, which is:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Given:
- [tex]\( (x_1, y_1) = (1, 2) \)[/tex]
- [tex]\( (x_2, y_2) = (16, p) \)[/tex]
- Distance = 17
Substitute the given values into the distance formula:
[tex]\[ 17 = \sqrt{(16 - 1)^2 + (p - 2)^2} \][/tex]
Simplify inside the square root:
[tex]\[ 17 = \sqrt{15^2 + (p - 2)^2} \][/tex]
Calculate [tex]\( 15^2 \)[/tex]:
[tex]\[ 15^2 = 225 \][/tex]
Substitute back into the equation:
[tex]\[ 17 = \sqrt{225 + (p - 2)^2} \][/tex]
Square both sides to eliminate the square root:
[tex]\[ 17^2 = 225 + (p - 2)^2 \][/tex]
Calculate [tex]\( 17^2 \)[/tex]:
[tex]\[ 17^2 = 289 \][/tex]
So, we have:
[tex]\[ 289 = 225 + (p - 2)^2 \][/tex]
Isolate [tex]\((p - 2)^2\)[/tex]:
[tex]\[ 289 - 225 = (p - 2)^2 \][/tex]
[tex]\[ 64 = (p - 2)^2 \][/tex]
Take the square root of both sides:
[tex]\[ \sqrt{64} = |p - 2| \][/tex]
Calculate the square root:
[tex]\[ 8 = |p - 2| \][/tex]
This gives us two possible solutions:
[tex]\[ p - 2 = 8 \quad \text{or} \quad p - 2 = -8 \][/tex]
Solve for [tex]\( p \)[/tex] in each case:
1. [tex]\( p - 2 = 8 \)[/tex]
[tex]\[ p = 8 + 2 \][/tex]
[tex]\[ p = 10 \][/tex]
2. [tex]\( p - 2 = -8 \)[/tex]
[tex]\[ p = -8 + 2 \][/tex]
[tex]\[ p = -6 \][/tex]
Therefore, the value of [tex]\( p \)[/tex] can be [tex]\( 10 \)[/tex] or [tex]\( -6 \)[/tex].
