Question

point c is \\(\frac{3}{4}\\) of the way from point a(-4, -2) to point b(8, 6). what are the coordinates of c?
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Explanation:

Step1: Recall the section formula

The coordinates of a point \( C(x,y) \) that divides the line segment joining \( A(x_1,y_1) \) and \( B(x_2,y_2) \) in the ratio \( m:n \) are given by \( x=\frac{mx_2 + nx_1}{m + n} \) and \( y=\frac{my_2+ny_1}{m + n} \). Here, since \( C \) is \( \frac{3}{4} \) of the way from \( A \) to \( B \), the ratio \( m:n = 3:1 \) (because the remaining part is \( 1-\frac{3}{4}=\frac{1}{4} \), so the ratio of the part from \( A \) to \( C \) and \( C \) to \( B \) is \( 3:1 \)). Given \( A(-4,-2) \) so \( x_1=-4,y_1 = - 2 \) and \( B(8,6) \) so \( x_2 = 8,y_2=6 \), \( m = 3,n = 1 \).

Step2: Calculate the x - coordinate of C

Using the formula for \( x \):
\( x=\frac{3\times8+1\times(-4)}{3 + 1}=\frac{24-4}{4}=\frac{20}{4}=5 \)

Step3: Calculate the y - coordinate of C

Using the formula for \( y \):
\( y=\frac{3\times6+1\times(-2)}{3 + 1}=\frac{18 - 2}{4}=\frac{16}{4}=4 \)

Answer:

The coordinates of point \( C \) are \( (5,4) \)