Triangle LMN hes vertices at L(-4,-3), M(2,5), and N(-13,10). If the distance from point P(x,-2) to L equals the distance from P to M, what is the value of x?
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To find the value of x that satisfies the given condition, we first need to calculate the distances between point P and points L and M.
Distance Formula: The distance between two points (x1, y1) and (x2, y2) in a coordinate plane is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the distance between points P and L:
d_PL = sqrt((x - (-4))^2 + (-2 - (-3))^2) = sqrt((x + 4)^2 + 1)
Now, let's calculate the distance between points P and M:
d_PM = sqrt((x - 2)^2 + (-2 - 5)^2) = sqrt((x - 2)^2 + 49)
According to the given condition, the distance from P to L should be equal to the distance from P to M:
sqrt((x + 4)^2 + 1) = sqrt((x - 2)^2 + 49)
To solve this equation, we need to square both sides to eliminate the square roots:
(x + 4)^2 + 1 = (x - 2)^2 + 49
Expand both sides of the equation:
x^2 + 8x + 16 + 1 = x^2 - 4x + 4 + 49
Combine like terms:
x^2 - x^2 + 8x + 4x = 49 - 16 - 4 - 1
Simplify:
12x = 28
Divide both sides by 12:
x = 28 / 12
Simplify:
x = 7/3
Therefore, the value of x that satisfies the given condition is 7/3.