Question

the coordinates of the endpoints of \\(\overline{st}\\) are \\(s(3, 5)\\) and \\(t(10, 19)\\). point \\(u\\) is on \\(\overline{st}\\) and divides it such that \\(su:tu\\) is \\(1:6\\). what are the coordinates of \\(u\\)? write your answers as integers or decimals. \\((\square, \square)\\) submit

Explanation:

Step1: Usar la fórmula de división de segmento

La fórmula para encontrar el punto \( U(x,y) \) que divide el segmento de \( S(x_1,y_1) \) a \( T(x_2,y_2) \) en la razón \( m:n \) es:
\[
x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}
\]
Aquí, \( S(3,5) \), \( T(10,19) \), \( m = 1 \), \( n = 6 \).

Step2: Calcular la coordenada x de U

Sustituir \( x_1 = 3 \), \( x_2 = 10 \), \( m = 1 \), \( n = 6 \) en la fórmula de x:
\[
x = \frac{1 \times 10 + 6 \times 3}{1 + 6} = \frac{10 + 18}{7} = \frac{28}{7} = 4
\]

Step3: Calcular la coordenada y de U

Sustituir \( y_1 = 5 \), \( y_2 = 19 \), \( m = 1 \), \( n = 6 \) en la fórmula de y:
\[
y = \frac{1 \times 19 + 6 \times 5}{1 + 6} = \frac{19 + 30}{7} = \frac{49}{7} = 7
\]

Answer:

(4, 7)