Answer:
x = 15 or 3
Step-by-step explanation:
To find the length of a segment given its endpoints on the coordinate plane, we can use 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] \Large\boxed{\boxed{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} }}[/tex]
Given that one endpoint is [tex] (9, 13) [/tex] and the other endpoint is [tex] (x, -2) [/tex], and the length of the segment is [tex] 3\sqrt{29} [/tex] units, we can write the distance formula as follows:
[tex] 3\sqrt{29} = \sqrt{(x - 9)^2 + (-2 - 13)^2} [/tex]
[tex] 3\sqrt{29} = \sqrt{(x - 9)^2 + (-15)^2} [/tex]
[tex] 3\sqrt{29} = \sqrt{(x - 9)^2 + 225} [/tex]
[tex] (3\sqrt{29})^2 = (x - 9)^2 + 225 [/tex]
[tex] 9 \times 29 = (x - 9)^2 + 225 [/tex]
[tex] 261 = (x - 9)^2 + 225 [/tex]
[tex] 261 - 225 = (x - 9)^2 [/tex]
[tex] 36 = (x - 9)^2 [/tex]
Now, we take the square root of both sides:
[tex] \sqrt{36} = \sqrt{(x - 9)^2} [/tex]
[tex] 6 = |x - 9| [/tex]
This gives us two possible equations:
[tex] x - 9 = 6 [/tex]
[tex] x - 9 = -6 [/tex]
Solving each equation separately:
First one:
[tex] x - 9 = 6 [/tex]
[tex] x = 6 + 9 = 15 [/tex]
Second one;
[tex] x - 9 = -6 [/tex]
[tex] x = -6 + 9 = 3 [/tex]
Therefore, the possible values of [tex] x [/tex] are [tex] 15 [/tex] and [tex] 3 [/tex].
