If AB=5x+2 and AC=12x-2, what is BC?
To find the length of segment BC, we can use the distance formula:
The distance formula states that for two points (x₁, y₁) and (x₂, y₂) in a coordinate plane, the distance between them is given by:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
In this case, we only have the x-coordinates of points A, B, and C. Therefore, we can represent the points as (x₁, 0) for A, (x₂, 0) for B, and (x₃, 0) for C.
Let's calculate the distance BC using the distance formula:
BC = √((x₃ - x₂)² + (0 - 0)²)
Given AB = 5x + 2 and AC = 12x - 2, we need to find the values of x that correspond to points B and C.
Since AB = BC + AC, we can set up the equation:
BC + AC = 5x + 2 + 12x - 2
Simplifying, we get:
BC + AC = 17x
Since BC = √((x₃ - x₂)² + (0 - 0)²), we can rewrite the equation as:
√((x₃ - x₂)² + (0 - 0)²) + 12x - 2 = 17x
Simplifying further, we have:
(x₃ - x₂)² = (17x - 12x + 2)²
Simplifying the right side, we get:
(x₃ - x₂)² = (5x + 2)²
Taking the square root of both sides, we have:
x₃ - x₂ = √((5x + 2)²)
Now, we can rearrange the equation to isolate BC:
BC = x₃ - x₂
Substituting the value we obtained above, we get:
BC = √((5x + 2)²)
Therefore, the length of segment BC is given by √((5x + 2)²).