Final answer:
To find the y-coordinate of a triangle's vertex with a given x-coordinate, point of concurrency, and distance from it to vertices, we use the distance formula, resulting in two possible y-values.
Step-by-step explanation:
The question involves finding the y-coordinate of a vertex of a triangle when its x-coordinate is given, the distance from the point of concurrency of the perpendicular bisectors is known, and the point of concurrency is also known. Given that the point of concurrency is at (9,1) and the distance to each vertex is 13 units, we can use the distance formula to find the y-coordinate of the vertex with x-coordinate 21.
The distance formula is given by the equation √((x2 - x1)² + (y2 - y1)²) = distance. Substituting the known values, we get 13 = √((21 - 9)² + (y - 1)²). Simplifying, we find that (y - 1)² = 13² - (21 - 9)², which results in two possible y-values upon solving.
By calculating we discover the y-values are 5 and -3. Therefore, the y-coordinate of the vertex must be either 5 or -3, alongside the x-coordinate of 21.