find x such that (x,7) is 10 units from (-4,1)
10^2=(x+4)^2 + (7-1)^2 that is the distance formula. Solve for x.
To find the value of x such that (x, 7) is 10 units away from (-4, 1), we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by:
√((x2 - x1)^2 + (y2 - y1)^2) = Distance
In this case, the distance is given as 10 units and the two points are (-4, 1) and (x, 7). By substituting the given values into the distance formula, we can solve for x.
√((x - (-4))^2 + (7 - 1)^2) = 10
Simplifying:
√((x + 4)^2 + 6^2) = 10
Squaring both sides:
(x + 4)^2 + 6^2 = 10^2
(x + 4)^2 + 36 = 100
(x + 4)^2 = 100 - 36
(x + 4)^2 = 64
Taking the square root of both sides:
x + 4 = ±√64
x + 4 = ±8
Solving for x in both cases:
Case 1: x + 4 = 8
x = 8 - 4
x = 4
Case 2: x + 4 = -8
x = -8 - 4
x = -12
Therefore, the two values of x such that (x, 7) is 10 units away from (-4, 1) are x = 4 and x = -12.
To find the value of x such that (x,7) is 10 units from (-4,1), we can use the distance formula.
The distance formula is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, let (x₁, y₁) = (-4, 1) and (x₂, y₂) = (x, 7).
Plugging the values into the formula, we have:
10 = √((x - (-4))² + (7 - 1)²)
Simplifying further:
100 = (x + 4)² + 36
Expand the equation:
100 = x² + 8x + 16 + 36
Combine like terms:
100 = x² + 8x + 52
Rearrange the equation to find the value of x:
x² + 8x + 52 - 100 = 0
x² + 8x - 48 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring this equation is a bit challenging, so let's use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = 8, and c = -48. Substituting these values into the formula:
x = (-8 ± √(8² - 4(1)(-48))) / (2 * 1)
Simplifying further:
x = (-8 ± √(64 + 192)) / 2
x = (-8 ± √256) / 2
x = (-8 ± 16) / 2
We have two possible solutions:
1. x = (-8 + 16) / 2 = 8 / 2 = 4
2. x = (-8 - 16) / 2 = -24 / 2 = -12
Therefore, the values of x such that (x,7) is 10 units from (-4,1) are x = 4 and x = -12.